Positivity vs negativity of canonical bases
aa r X i v : . [ m a t h . R T ] F e b POSITIVITY VS NEGATIVITY OF CANONICAL BASES
YIQIANG LI AND WEIQIANG WANG
Abstract.
We provide examples for negativity of structure constants of the stably canon-ical basis of modified quantum gl n and an analogous basis of modified quantum coideal al-gebra of gl n . In contrast, we construct the canonical basis of the modified quantum coidealalgebra of sl n , establish the positivity of its structure constants, the positivity with respectto a geometric bilinear form as well as the positivity of its action on the tensor powers ofthe natural representation. The matrix coefficients of the transfer map on the associatedSchur algebras with respect to the canonical bases are shown to be positive. Formulas forcanonical basis of the Schur algebra of rank one are obtained. Dedicated to George Lusztig for his 70th birthday with admiration
Contents
1. Introduction 12. Negativity of the stably canonical basis of ˙ U ( gl n ) 43. Positivity of canonical basis of ˙ U ( sl n ) and a basis of ˙ U ( gl n ) 94. Modified quantum coideal algebras ˙ U ( gl n ) and ˙ U ( sl n ), for n odd 135. Canonical basis for modified quantum coideal algebra ˙ U ( sl n ), for n odd 186. Canonical basis for ˙ U ı ( sl n ) for n even 257. Formulas of canonical basis of S (2 , d ) 32References 371. Introduction S ( n, d ) geometrically in terms of pairs of partial flags of type A . Furthermore, they con-struct the modified quantum group ˙ U ( gl n ) via a stabilization procedure from the family ofalgebras S ( n, d ) as d varies. The IC construction provides a canonical basis for S ( n, d ) whosestructure constants are positive (i.e., in N [ v, v − ]), which in turn via stabilization leads to adistinguished bar-invariant basis (which we shall refer to as BLM or stably canonical basis )for ˙ U ( gl n ).Recently the constructions of [BLM90] have been generalized to partial flag varieties oftype B and C in [BKLW] (also see [FL14] for type D ). A family of iSchur algebras iS ( n, d )was realized geometrically together with canonical (=IC) bases whose structure constants liein N [ v, v − ]. Via a stabilization procedure these algebras give rise to a limit algebra whichwas shown to be isomorphic to the modified quantum coideal algebra i ˙ U ( gl n ) of gl n , andwhich also admits a stably canonical basis. The appearance of the quantum coideal algebrawas inspired by [BW13] where a new approach to Kazhdan-Lusztig theory of type B/C via a new theory of canonical bases arising from quantum coideal algebras was developed. Eventhough the constructions for n odd and even are quite different with the case of even n beingmore challenging [BW13], one can carry out the construction in the even n case by relatingto the odd n case via a more subtle two-step stabilization [BKLW].1.2. The original motivation of this paper is to understand the positivity of the stablycanonical basis of the modified quantum coideal algebra i ˙ U ( gl n ). To that end, we haveto understand first the same positivity issue for ˙ U ( gl n ), as ˙ U ( gl ⌊ n ⌋ ) is simpler and also itappears essentially as a subalgebra of i ˙ U ( gl n ) with compatible stably canonical bases. Thecanonical bases arising from quantum groups of ADE type are widely expected to enjoy allkinds of positivity (see [L90, L93]), and there is no indication in the literature that anythingon ˙ U ( gl n ) (or gl n ) differs substantially from its counterpart on ˙ U ( sl n ) (or sl n ).To our surprise, the behavior of the BLM/stably canonical basis of ˙ U ( gl n ) turns out to bedramatically different, already for n = 2, from the canonical basis of ˙ U ( sl n ). In particular, weprovide examples that the structure constants of the stably canonical basis are negative, andthat the stably canonical basis of ˙ U ( gl n ) fails to descent to the canonical basis of the finite-dimensional simple ˙ U ( gl n )-modules. These examples, though not difficult, are unexpectedamong the experts whom we have a chance to communicate with, so we write them downhoping to clarify some confusion or false expectation. The fundamental reason behind thefailure of positivity of the BLM basis and beyond is that the stabilization process is notentirely geometric (when the involved matrices contain negative diagonal entries).The structure constants of the canonical basis of ˙ U ( sl n ) are positive; this follows easilyfrom combining the positivity of the canonical (=IC) basis of the Schur algebras [BLM90]with a result of McGerty [M12, Proposition 7.8] (or with a stronger result of [SV00], whichconfirmed Lusztig’s conjectures [L99, Conjectures 9.2, 9.3]). For the reader’s convenience,we make explicit this positivity in Proposition 3.1 and supply a short proof, as it could notbe explicitly found in these earlier papers.1.3. Now we focus on the modified quantum coideal algebra i ˙ U ( sl n ), for n ≥
2. We con-struct a canonical basis for the modified quantum coideal algebra i ˙ U ( sl n ) which shares manyremarkable properties of the canonical basis for ˙ U ( sl n ). In particular, it has positive struc-ture constants, and it is characterized up to sign by the three properties: bar-invariance,integrality, and almost orthonormality with respect to a bilinear form of geometric origin.Moreover, it admits positivity with respect to the geometric bilinear form. In addition, thiscanonical basis is compatible with Lusztig’s under a natural inclusion ˙ U ( sl ⌊ n ⌋ ) ⊆ i ˙ U ( sl n ).Our argument largely follows the line in McGerty’s work [M12] for n odd (the case for n even needs substantial new work), though we have avoided using the non-degeneracy ofthe geometric bilinear form of i ˙ U ( sl n ), which was not available at the outset. Instead, thenon-degeneracy of the bilinear form is replaced by arguments involving the stably canonicalbasis of i ˙ U ( gl n ) from [BKLW] and the non-degeneracy eventually follows from the almostorthonormality of the canonical basis which we establish.We further show that the transfer map on the iSchur algebras sends every canonical basiselement to a positive sum of canonical basis elements or zero. Some basic properties on thetransfer map established in [FL15] are used here. Moreover, the matrix coefficients (withrespect to canonical basis) for the action of any canonical basis element in i ˙ U ( sl n ) on V ⊗ d OSITIVITY VS NEGATIVITY OF CANONICAL BASES 3 are shown to be positive, where V is the n -dimensional natural representation of i ˙ U ( sl n ).We remark that the transfer maps on the type A Schur algebras were earlier studied in[L99, L00, SV00, M12].As in [BW13, BKLW], the different behaviors in the cases for n odd and even force us tocarry out the studies of the two cases separately in this paper. The case of odd n , indicatedby the superscript , is easier and done first, while the remaining case is indicated by thesuperscript ı . Let us set up some notations used in the main text. For n odd and hence n = n − U ( gl n ) = i ˙ U ( gl n ), S ( n, d ) = iS ( n, d ), ˙ U ı ( gl n ) = i ˙ U ( gl n ),and S ı ( n , d ) = iS ( n , d ).There is another purely representation theoretic approach in [BW16] toward the bilinearforms and canonical bases for general quantum coideal algebras including i ˙ U ( sl n ), whichnevertheless cannot address the positivity of canonical bases. Note that the papers [L99,L00, SV00, M12] are mostly concerned about the quantum Schur algebras and quantumgroups of affine type A . A geometric setting for the quantum coideal algebras of affine typewill be pursued elsewhere.1.4. The paper is organized as follows. In Section 2, we construct examples that a naturalshift map (which is an algebra isomorphism) on ˙ U ( gl n ) does not preserve the BLM basis,that the structure constants of BLM basis for ˙ U ( gl n ) are negative, and that the BLM basisof ˙ U ( gl n ) does not descend to the canonical basis of a finite-dimensional simple module.In Section 3, we show that the positivity of structure constants for the canonical basis of˙ U ( sl n ) is an easy consequence of McGerty’s results. Then we construct a positive basis for˙ U ( gl n ) with positive structure constants by transporting the canonical basis of ˙ U ( sl n ). Weexplain several positivity results on the transfer map for Schur algebras.In Sections 4, 5, and 6, we study the quantum coideal algebras and the associated Schuralgebras. In Section 4, we show the stably canonical basis constructed in [BKLW] for themodified quantum coideal algebra ˙ U ( gl n ) for n odd does not have positive structure con-stants.In Section 5, we set n to be odd, and study the behavior of the canonical bases of the Schur algebras S ( n, d ) and varying d ≫ U ( sl n ). We show thatthe structure constants of the canonical basis of ˙ U ( sl n ) are positive. We further show thatthe transfer map sends every canonical basis element to a positive sum of canonical basiselements or zero.In Section 6, we treat S ı ( n , d ) and ˙ U ı ( sl n ) for n even, which is more subtle. We show thatthe main results in Section 5 can be obtained in this case as well though extra technical workis required.In Section 7, we present explicit formulas of the canonical basis of the rank one iSchuralgebra in terms of the standard basis elements. Some interesting combinatorial identitieswhich seem new are obtained along the way. Acknowledgements.
We are grateful to Huanchen Bao and Zhaobing Fan for relatedcollaborations and many stimulating discussions. We thank Olivier Schiffmann and Ben
YIQIANG LI AND WEIQIANG WANG
Webster for very helpful comments. The second author is partially supported by NSF DMS-1405131; he thanks the Institute of Mathematics, Academia Sinica (Taipei) and InstitutMittag-Leffler for an ideal working environment and support.2.
Negativity of the stably canonical basis of ˙ U ( gl n )In this section, we construct several examples which show that a natural shift map on˙ U ( gl n ) does not preserve the BLM basis, that the structure constants of BLM basis for˙ U ( gl n ) are negative, and that the BLM basis of ˙ U ( gl n ) does not descend to the canonicalbasis of a finite-dimensional simple modules.2.1. The BLM preliminaries.
We recall some basics from [BLM90] (also see [DDPW]).Let v be a formal parameter, and A = Z [ v, v − ]. Let F q be a finite field of order q . Let N = { , , , . . . } . Let A S ( n, d ) (denoted by K d in [BLM90]) be the quantum Schur algebraover A , which specializes at v = √ q to the convolution algebra of pairs of n -step partial flagsin F dq . The algebra A S ( n, d ) admits a bar involution, a standard basis [ A ], and a canonical(= IC) basis { A } parameterized byΘ d = n A = ( a ij ) ∈ Mat n × n ( N ) | | A | = d o , where | A | = P ≤ i,j ≤ n a ij . Set Θ := ∪ d ≥ Θ d . The multiplication formulas of the A -algebras A S ( n, d ) exhibit some remarkable stabilityas d varies, which leads to a “limit” A -algebra K . The bar involution on A S ( n, d ) induces abar involution on K . The algebra K has a standard basis [ A ] and a BLM (or stably canonical )basis { A } , parameterized by˜Θ = { A = ( a ij ) ∈ Mat n × n ( Z ) | a ij ≥ i = j ) } . Denote by ǫ i the i -th standard basis element in Z n . For 1 ≤ h ≤ n − a ≥ λ ∈ Z n , wedenote by E ( a ) h,h +1 ( λ ) the matrix whose ( h, h + 1)th entry is a , whose diagonal coincides with λ − aǫ h +1 , and all other entries are zero. Similarly, denote by E ( a ) h +1 ,h ( λ ) the matrix whose( h + 1 , h )th entry is a , whose diagonal coincides with λ − aǫ h , and all other entries are zero.Recall the A -form of the modified quantum gl n , denoted by A ˙ U ( gl n ), is generated by theidempotents 1 λ (for λ ∈ Z n ) and the divided powers E ( a ) i λ , F ( a ) i λ (for a ≥ ≤ i ≤ n − A -algebra isomorphism K ∼ = A ˙ U ( gl n ),which sends [ E ( a ) h,h +1 ( λ )] to E ( a ) h λ and [ E ( a ) h +1 ,h ( λ )] to F ( a ) h λ , for all admissible λ , h and a .We shall always make such an identification K ≡ A ˙ U ( gl n ) and use only A ˙ U ( gl n ) in theremainder of the paper.We denote S ( n, d ) = Q ( v ) ⊗ A A S ( n, d ) , ˙ U ( gl n ) = Q ( v ) ⊗ A A ˙ U ( gl n ) . The algebra ˙ U ( gl n ) is a direct sum of subalgebras:(2.1) ˙ U ( gl n ) = M d ∈ Z ˙ U ( gl n ) h d i , where ˙ U ( gl n ) h d i is spanned by elements of the form 1 λ u µ with | µ | = | λ | = d and u ∈ ˙ U ( gl n );here as usual we denote | λ | = λ + . . . + λ n , for λ = ( λ , . . . , λ n ) ∈ Z n . OSITIVITY VS NEGATIVITY OF CANONICAL BASES 5
The elements [ E ( a ) h,h +1 ( λ )] for E ( a ) h,h +1 ( λ ) ∈ Θ d and [ E ( a ) h +1 ,h ( λ )] for E ( a ) h +1 ,h ( λ ) ∈ Θ d (for alladmissible h, a, λ ) generate the A -algebra A S ( n, d ).Let 0 i,j be the i × j zero matrix. Fix two positive integers m, n such that m < n . Let k ∈ Z . By using the multiplication formulas in [BLM90, 4.6], we note that the assignment[ A ] (cid:20) A m,n − m n − m,m kI (cid:21) defines an algebra embedding ι km,n : A ˙ U ( gl m ) −→ A ˙ U ( gl n ) . The following lemma, which basically follows from the definition of the BLM basis, willbe used later on.
Lemma 2.1.
Let m, n, k ∈ Z with < m < n . Then ι km,n ( { A } ) = (cid:26) A m,n − m n − m,m kI (cid:27) forall A ∈ ˜Θ . Incompatibility of BLM bases under the shift map.
Given p ∈ Z , it follows fromthe multiplication formulas [BLM90, 4.6] that there exists an algebra isomorphism (called ashift map ) ξ p : ˙ U ( gl n ) −→ ˙ U ( gl n ) , ξ p ([ A ]) = [ A + pI ] , (2.2)for all A such that A is either diagonal, E h,h +1 ( λ ) or E h +1 ,h ( λ ) for some 1 ≤ h ≤ n − I denotes the identity matrix. Note that ξ p commutes with the bar involution and ξ p preservesthe A -form A ˙ U ( gl n ). Note also that ξ − p = ξ − p .Introduce the (not bar-invariant) quantum integers and quantum binomials, for m ∈ Z and b ∈ N ,(2.3) (cid:20) mb (cid:21) = (cid:20) mb (cid:21) v = Y ≤ i ≤ b v m − i +1) − v i − , and [ m ] = (cid:20) m (cid:21) = v m − v − . Lemma 2.2.
Let n = 2 . If a ≥ , a ≤ − and p ≤ , then (cid:26) p a a + p (cid:27) = (cid:20) p a a + p (cid:21) − v a +1 [ p + 1] (cid:20) p + 1 0 a − a + p + 1 (cid:21) . Proof.
We denote the multiplication in ˙ U ( gl ) by ∗ to avoid confusion with the usual matrixmultiplication. We will repeatedly use the fact that [ A ] is bar-invariant (divided powers) for A upper- or lower-triangular.The formula [BLM90, 4.6(a)] gives us (for all a , a ∈ Z and a ≥ (cid:20) a a + a (cid:21) ∗ (cid:20) a a a + 1 (cid:21) = (cid:20) a a a (cid:21) + v a − a − [ a + 1] (cid:20) a + 1 0 a − a + 1 (cid:21) . (2.4)By applying the bar map to (2.4) and then comparing with (2.4) again, we have (cid:20) a a a (cid:21) = (cid:20) a a a (cid:21) + (cid:16) v a − a − [ a + 1] − v − a + a +1 [ a + 1] (cid:17) (cid:20) a + 1 0 a − a + 1 (cid:21) . YIQIANG LI AND WEIQIANG WANG
By a change of variables we obtain that (for p ∈ Z ) (cid:20) p a a + p (cid:21) = (cid:20) p a a + p (cid:21) + (cid:16) v − a − [ p + 1] − v a +1 [ p + 1] (cid:17) (cid:20) p + 1 0 a − a + p + 1 (cid:21) . (2.5)Hence we can write (cid:26) p a a + p (cid:27) = (cid:20) p a a + p (cid:21) + x (cid:20) p + 1 0 a − a + p + 1 (cid:21) , for some x ∈ v − Z [ v − ] . It follows by this and (2.5) that x − ¯ x = v − a − [ p + 1] − v a +1 [ p + 1] . Using the assumption that a ≤ − p ≤
0, we have v a +1 [ p + 1] ∈ v − Z [ v − ] andhence x = − v a +1 [ p + 1]. The lemma follows. (cid:3) Proposition 2.3.
The shift map ξ p : ˙ U ( gl n ) → ˙ U ( gl n ) (for p = 0 ) does not always preservethe BLM basis, for n ≥ . More explicitly, for n = 2 , if a ≥ , a ≤ − and p < , then ξ p (cid:26) a a (cid:27) = (cid:26) p a a + p (cid:27) + (cid:16) v − a − [ p ] + v a +3 [ p ] (cid:17) (cid:26) p + 1 0 a − a + p + 1 (cid:27) . Proof.
We first verify the formula for n = 2. By applying (2.4) twice, we have ξ p (cid:20) a a a (cid:21) = (cid:20) a + p a a + p (cid:21) + v − a − a − [ p ] (cid:20) a + p + 1 0 a − a + p + 1 (cid:21) . (2.6)The formula in Lemma 2.2 specializes at p = 0 to be (cid:26) a a (cid:27) = (cid:20) a a (cid:21) − v a +1 (cid:20) a − a + 1 (cid:21) . Hence, using (2.6) we have ξ p (cid:26) a a (cid:27) = (cid:20) p a a + p (cid:21) + ( v − a − [ p ] − v a +1 ) (cid:26) p + 1 0 a − a + p + 1 (cid:27) , (2.7)which can be readily turned into the formula in the proposition by Lemma 2.2.If ξ p preserved the BLM basis, then we would have ξ p ( { A } ) = { A + pI } by definitions, forall A . Hence the formula for ξ p (cid:26) a a (cid:27) (with p <
0) together with the fact ξ − p = ξ − p shows that ξ p (for p = 0) does not preserve the BLM basis.The proposition for general n ≥ (cid:3) Remark 2.4.
It can be shown similarly that ξ p (cid:26) a a (cid:27) = (cid:26) p a a + p (cid:27) , if a ≥ , a ≤ − p > . Indeed precise formulas for both sides of this inequality can be obtained by (2.5) and (2.7).
Remark 2.5.
There exists a surjective algebra homomorphism Φ d : ˙ U ( gl n ) → S ( n, d ) whichsends [ A ] to [ A ] for A ∈ Θ d or to 0 otherwise. It was shown in [Fu14] that Φ d preserves thecanonical bases, sending { A } to { A } for A ∈ Θ d or to 0 otherwise. Making a gl n analogywith [L99, 9.3], one might modify the map Φ d to define a new algebra homomorphism Φ ′ d :˙ U ( gl n ) → S ( n, d ) as follows: for u ∈ ˙ U ( gl n ) h d − pn i with p ∈ Z , we let Φ ′ d ( u ) = Φ d ( ξ p ( u )); OSITIVITY VS NEGATIVITY OF CANONICAL BASES 7 also let Φ ′ d | ˙ U ( gl n ) h d ′ i = 0 unless d ′ ≡ d mod n . It follows by Proposition 2.3 and Remark 2.4that Φ ′ d : ˙ U ( gl n ) → S ( n, d ) does not preserve the canonical bases for general d and n .2.3. Negativity of BLM structure constants.Proposition 2.6.
The structure constants for the algebra ˙ U ( gl n ) with respect to the BLMbasis are not always positive, for n ≥ . More explicitly, for n = 2 , we have (cid:26) − (cid:27) ∗ (cid:26) − (cid:27) = ( v + v − ) (cid:26) − − (cid:27) − (2 v − + 1 + 2 v ) (cid:26) − (cid:27) − ( v − + v − + 2 + v + v ) (cid:26) − (cid:27) . Proof.
It suffices to check the example for n = 2 in view of Lemma 2.1. We will repeatedlyuse the fact that [ A ] is bar-invariant (divided powers) for A upper- or lower-triangular.We claim the following identities hold: (cid:26) − (cid:27) = (cid:20) − (cid:21) − v − (cid:20) − (cid:21) , (2.8) (cid:26) − − (cid:27) = (cid:20) − − (cid:21) , (cid:26) − (cid:27) = (cid:20) − (cid:21) . (2.9)Indeed, (2.8) follows by Lemma 2.2, and the second identity of (2.9) is clear. Moreover, by[BLM90, 4.6(b)] and (2.8), we have (cid:20) − − (cid:21) = (cid:20) − (cid:21) ∗ (cid:20) − (cid:21) + ( v − + 1 + v ) (cid:20) − (cid:21) − ( v − + v − + 1) (cid:20) − (cid:21) = (cid:20) − (cid:21) ∗ (cid:20) − (cid:21) + ( v − + 1 + v ) (cid:26) − (cid:27) , which is bar invariant. Hence it must be a BLM basis element, whence (2.9).By [BLM90, 4.6(a),(b)] (also see (2.4)), we have (cid:20) − (cid:21) = (cid:20) − (cid:21) ∗ (cid:20) − (cid:21) − v (cid:20) − (cid:21) , (2.10) (cid:20) − (cid:21) ∗ (cid:20) − (cid:21) = ( v + v − ) (cid:20) − − (cid:21) − (1 + v ) (cid:20) − (cid:21) , (2.11) (cid:20) − (cid:21) ∗ (cid:20) − − (cid:21) = ( v + v − ) (cid:20) − − (cid:21) , (2.12) (cid:20) − (cid:21) ∗ (cid:20) − (cid:21) = (cid:20) − (cid:21) + v (cid:20) − (cid:21) . (2.13) YIQIANG LI AND WEIQIANG WANG
Therefore we have (cid:26) − (cid:27) ∗ (cid:26) − (cid:27) = (cid:20) − (cid:21) ∗ (cid:20) − (cid:21) ∗ (cid:20) − (cid:21) − v − (cid:20) − (cid:21) ∗ (cid:20) − (cid:21) − ( v + v − ) (cid:20) − (cid:21) + v − ( v + v − ) (cid:20) − (cid:21) = ( v + v − ) (cid:20) − − (cid:21) − (2 v − + 1 + 2 v ) (cid:20) − (cid:21) + ( v − − v − v ) (cid:20) − (cid:21) , (2.14)where the first identity above uses (2.8) and (2.10), while the second identity above uses(2.11), (2.12) and (2.13).With the help of (2.8) and (2.9), a direct computation shows the right-hand side of thedesired identity in the proposition is also equal to (2.14). The proposition is proved. (cid:3) Incompatability of BLM bases for ˙ U and L ( λ ) . Denote by L ( λ ) the ˙ U ( gl n )-moduleof highest weight λ with a highest weight vector u + λ . Proposition 2.7.
There exists a dominant integral weight λ and some BLM basis element C ∈ ˙ U ( gl n ) (for n ≥ ) such that Cu + λ is not a canonical basis element of L ( λ ) . Moreexplicitly, for n = 2 , if a ≥ , a ≤ − and p ≤ , λ = ( p + a , a + p + 1) , then (cid:26) p a a + p (cid:27) u + λ = v a +2 p +3 [ − a − p − F ( a − u + λ . Proof.
It suffices to verify such an example for n = 2 by using Lemma 2.1 where k is chosensuch that k ≤ a + p + 1.By [BLM90, 4.6], we have (cid:20) p + 1 0 a a + p (cid:21) ∗ (cid:20) p + a a + p (cid:21) = (cid:20) p a a + p (cid:21) + v a − [ a + p + 1] (cid:20) p + 1 0 a − a + p + 1 (cid:21) . By plugging the above equation into the formula in Lemma 2.2 (the assumption of which issatisfied), we obtain that (cid:26) p a a + p (cid:27) = (cid:20) p + 1 0 a a + p (cid:21) ∗ (cid:20) p + a a + p (cid:21) + v a +2 p +3 [ − a − p − (cid:20) p + 1 0 a − a + p + 1 (cid:21) , where we have used the identity − v a +1 [ p + 1] − v a − [ a + p + 1] = v a +2 p +3 [ − a − p − OSITIVITY VS NEGATIVITY OF CANONICAL BASES 9
Consider the dominant integral weight λ = ( p + a , a + p + 1). We have (cid:26) p a a + p (cid:27) u + λ = v a +2 p +3 [ − a − p − (cid:20) p + 1 0 a − a + p + 1 (cid:21) u + λ = v a +2 p +3 [ − a − p − F ( a − u + λ , which is not a canonical basis element in L ( λ ) if − a − p − > (cid:3) Remark 2.8.
It is shown in [Fu14, Proposition 4.7] that the BLM basis descends to thecanonical basis of L ( λ ) when the dominant highest weight λ is assumed to be in Z n ≥ .3. Positivity of canonical basis of ˙ U ( sl n ) and a basis of ˙ U ( gl n )In this section we exhibit various kinds of positivity of the canonical basis of ˙ U ( sl n ) andSchur algebras in relation to the transfer maps, most of which were known by experts thoughprobably in some other ways. We also construct a positive basis for ˙ U ( gl n ) by transportingthe canonical basis of ˙ U ( sl n ) to ˙ U ( gl n ).3.1. The algebras ˙ U ( gl n ) vs ˙ U ( sl n ) . We identify the weight lattice for gl n as Z n (regardedas the set of integral diagonal n × n matrices in ˜Θ if we think in the setting of K ), andwe define an equivalence ∼ on Z n by letting µ ∼ ν if and only if µ − ν = k (1 , . . . ,
1) forsome k ∈ Z . Denote by µ the equivalence class of µ ∈ Z n , and we identify the set of theseequivalence classes ¯ Z n as the weight lattice of sl n . We denote by | µ | ∈ Z /n Z the congruenceclass of | µ | modulo n . For later use we also extend this definition to define an equivalencerelation ∼ on ˜Θ: A ∼ A ′ if and only if A − A ′ = kI for some k ∈ Z . We set(3.1) Θ n = ˜Θ / ∼ . As a variant of ˙ U ( gl n ), the modified quantum group ˙ U ( sl n ) admits a family of idempotents1 µ , for µ ∈ ¯ Z n . The algebra ˙ U ( sl n ) is naturally a direct sum of n subalgebras:(3.2) ˙ U ( sl n ) = M ¯ d ∈ Z /n Z ˙ U ( sl n ) h ¯ d i , where ˙ U ( sl n ) h ¯ d i is spanned by 1 µ ˙ U ( sl n )1 λ , where | µ | ≡ | λ | ≡ d mod n . It follows that A ˙ U ( sl n ) = ⊕ ¯ d ∈ Z /n Z A ˙ U ( sl n ) h ¯ d i . We denote by π ¯ d : ˙ U ( sl n ) → ˙ U ( sl n ) h ¯ d i the projection to the¯ d -th summand.There exists a natural algebra isomorphism(3.3) ℘ d : ˙ U ( gl n ) h d i ∼ = ˙ U ( sl n ) h d i ( ∀ d ∈ Z ) , which sends 1 λ , E i λ and F i λ to 1 ¯ λ , E i λ and F i λ respectively, for all r , i , and all λ with | λ | = d . This induces an isomorphism ℘ λ : ˙ U ( gl n )1 λ ∼ = ˙ U ( sl n )1 λ , for each λ ∈ Z n ,and also an isomorphism µ ℘ λ : 1 µ ˙ U ( gl n )1 λ ∼ = 1 µ ˙ U ( sl n )1 λ , for all λ, µ ∈ Z n with | λ | = | µ | .(These isomorphisms further induce similar isomorphisms for the corresponding A -forms,which match the divided powers.) Combining ℘ d for all d ∈ Z gives us a homomorphism ℘ : ˙ U ( gl n ) → ˙ U ( sl n ). It follows by definitions that(3.4) ℘ ◦ ξ p = ℘, for all p ∈ Z . Recall from Remark 2.5 the surjective algebra homomorphism Φ d : ˙ U ( gl n ) → S ( n, d ). Thealgebra homomorphism φ d : ˙ U ( sl n ) → S ( n, d ) is defined as the composition(3.5) φ d : ˙ U ( sl n ) π ¯ d −→ ˙ U ( sl n ) h ¯ d i ℘ d −→ ˙ U ( gl n ) h d i Φ d −→ S ( n, d ) . It follows that φ d | ˙U ( sl n ) h ¯ d ′ i = 0 if ¯ d ′ = ¯ d , and we have a surjective homomorphism φ d :˙ U ( sl n ) h ¯ d i → S ( n, d ). Clearly φ d preserves the A -forms.3.2. Positivity of canonical basis for ˙ U ( sl n ) . The canonical basis of A ˙ U ( sl n ) (and henceof ˙ U ( sl n )) is defined by Lusztig [L93], and it is further studied from a geometric viewpointby McGerty [M12]. The following positivity for canonical basis could (and probably should)have been formulated explicitly in [M12], as there is no difficulty to establish it therein.Given an n × n matrix A , we shall denote p A = A + pI, where I is the identity matrix. Proposition 3.1.
The structure constants of the canonical basis for the algebra ˙ U ( sl n ) liein N [ v, v − ] , for n ≥ .Proof. Let ˙ B ( sl n ) = ∪ ¯ d ∈ Z /n Z ˙ B ( sl n ) h ¯ d i be the canonical basis for ˙ U ( sl n ), where ˙ B ( sl n ) h ¯ d i isa canonical basis for ˙ U ( sl n ) h ¯ d i . Let a, b ∈ ˙ B ( sl n ) h ¯ d i , for some ¯ d . We have, for some suitablefinite subset Ω ⊂ ˙ B ( sl n ) h ¯ d i ,(3.6) a ∗ b = X z ∈ Ω P za,b z. It is shown [M12] that there exists a positive integer d in the congruence class ¯ d and A, B, C z ∈ Θ d such that φ d + pn ( a ) = { p A } , φ d + pn ( b ) = { p B } , φ d + pn ( z ) = { p C z } , for all p ≫ φ d + pn to (3.6) we have { p A } ∗ { p B } = X z ∈ Ω P za,b { p C z } . The structure constants for the canonical basis of the Schur algebra S ( n, d + pn ) are wellknown to be in N [ v, v − ] thanks to the intersection cohomology construction [BLM90], andhence P za,b ∈ N [ v, v − ].Since the algebra ˙ U ( sl n ) is a direct sum of the algebras ˙ U ( sl n ) h ¯ d i for ¯ d ∈ Z /n Z , theproposition is proved. (cid:3) Remark 3.2.
The positivity as in Proposition 3.1 was conjectured by Lusztig [L93] formodified quantum group of symmetric type. There is a completely different proof of such apositivity in ADE type via categorification technique by Webster [Web]. The argument herealso shows the positivity of the canonical basis of modified quantum affine sl n , based againon McGerty’s work. OSITIVITY VS NEGATIVITY OF CANONICAL BASES 11
Transfer map and positivity.
The transfer map for the v -Schur algebras φ d + n,d : A S ( n, d + n ) −→ A S ( n, d ) , or φ d + n,d : S ( n, d + n ) → S ( n, d ) by a base change, was defined geometrically by Lusztig[L00] and can also be described algebraically as follows. Set E i ; d = P λ [ E i,i +1 ( λ )] summedover all E i,i +1 ( λ ) ∈ Θ d , F i ; d = P λ [ E i +1 ,i ( λ )] summed over all E i +1 ,i ( λ ) ∈ Θ d , and K a ; d = P b ∈ N n , | b | = d v a · b b . (Here a · b = P i a i b i for a = ( a , . . . , a n ).) Then S ( n, d ) is generated bythese elements (see [BLM90]), and the transfer map φ d + n,d is characterized by φ d + n,d ( E i ; d + n ) = E i ; d , φ d + n,d ( F i ; d + n ) = F i ; d , φ d + n,d ( K a ; d + n ) = v | a | K a ; d . Recall the homomorphism φ d : ˙ U ( sl n ) → S ( n, d ) from (3.5). We have the followingcommutative diagram by matching the Chevalley generators (see [L99, L00]):˙ U ( sl n ) φ d + n v v ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ φ d ( ( PPPPPPPPPPPPP S ( n, d + n ) φ d + n,d / / S ( n, d )(3.7) Proposition 3.3.
The transfer map φ d + n,d : S ( n, d + n ) −→ S ( n, d ) sends each canonical ba-sis element to a sum of canonical basis elements with (bar invariant) coefficients in N [ v, v − ] or zero.Proof. Recall that φ d + n,d is the composition ( ξ ⊗ χ )∆, where ξ and ∆ are defined in [L00, 2.2,2.3]. The positivity of ξ with respect to the canonical bases is clear from the definition (asit is just a rescaling operator by some v -powers depending on the weights). The positivityof ∆ with respect to the canonical bases follows by its well-known identification with (thefunction version of) a hyperbolic localization functor and then appealing to the main theoremof Braden [Br03].So it suffices to show the positivity of the homomorphism χ : S ( n, n ) −→ Q ( v ). Recallthat the function χ is defined by χ ([ A ]) = v − d A det( A ) where d A = P i ≥ k,j 1. It is straightforward to check by [BLM90, Lemma 3.8] that { s i } = [ s i ]+ v − [ I ].Hence χ ( { s i } ) = v − det s i + v − det I = 0. Let w ∈ S n with ℓ ( w ) > 1. We can find an s i such that w = s i w ′ with ℓ ( w ′ ) + 1 = ℓ ( w ). By the construction of the Kazhdan-Lusztigbasis [KL79, § { s i } ∗ { w ′ } = { w } + X x : ℓ ( x ) <ℓ ( w ′ ) ,ℓ ( s i x ) <ℓ ( x ) µ ( x, w ′ ) { x } , µ ( x, w ′ ) ∈ A . (Note the x in the summation satisfies x = I .) Now applying the algebra homomorphism χ to the above identity and using the induction hypothesis, we see that χ ( { w } ) = 0. Thisfinishes the proof of the claim and hence of the theorem. (cid:3) Proposition 3.4. The map φ d : ˙ U ( sl n ) → S ( n, d ) sends each canonical basis element to asum of canonical basis elements with (bar invariant) coefficients in N [ v, v − ] or zero.Proof. Let b ∈ ˙ B ( sl n ). We can assume that b ∈ ˙ B ( sl n ) h ¯ d i as otherwise we have φ d ( b ) = 0.By [M12, Corollary 7.6, Proposition 7.8], φ d + pn ( b ) is a canonical basis element in S ( n, d + pn ),for some p ≫ 0. Using the commutative diagram (3.7) repeatedly, we have φ d ( b ) = φ d + n,d φ d +2 n,d + n · · · φ d + pn,d + pn − n (cid:0) φ d + pn ( b ) (cid:1) . It follows by repeatedly applying Proposition 3.3 that the term on the right-hand side aboveis a sum of canonical basis elements in S ( n, d ) with coefficients in N [ v, v − ]. (cid:3) Remark 3.5. Proposition 3.3 is partly inspired by [M12, Remark 7.10], and probably it canalso be proved by a possible functor realization of the transfer map, whose existence washinted at loc. cit. Note that stronger versions of Propositions 3.3 and 3.4 hold (which statethat the canonical bases are preserved by φ d + n,d and φ d ), according to the main results of[SV00] (which proved Lusztig’s conjectures [L99]). Our short yet transparent proofs of theweaker statements above might be of interest to the reader, and they will be adapted in latersections to the modified quantum coideal algebras and their associated Schur algebras.Recall [GL92] that the Schur-Jimbo ( S ( n, d ) , H S d )-duality on V ⊗ d can be realized geomet-rically, where V is n -dimensional and H S d is the Iwahori-Hecke algebra associated to thesymmetric group S d . Denote by B ( n d ) the canonical basis of V ⊗ d . The canonical bases on V ⊗ d as well as on S ( n, d ) are realized as simple perverse sheaves, and the action of S ( n, d )on V ⊗ d is realized in terms of a convolution product. Hence we have the following positivity. Proposition 3.6. [GL92] The action of S ( n, d ) on V ⊗ d with respect to the correspondingcanonical bases is positive in the following sense: for any canonical basis element a of S ( n, d ) and any b ∈ B ( n d ) , we have a ∗ b = X b ′ ∈ B ( n d ) C b ′ a,b b ′ , where C b ′ a,b ∈ N [ v, v − ] . We shall take the liberty of saying some action is positive in different contexts similar tothe above proposition. Now that ˙ U ( sl n ) acts on V ⊗ d naturally by composing the action of S ( n, d ) on V ⊗ d with the map φ d : ˙ U ( sl n ) → S ( n, d ). We have the following corollary ofPropositions 3.4 and 3.6. Corollary 3.7. The action of ˙ U ( sl n ) on V ⊗ d with respect to the corresponding canonicalbases is positive. Note by [L93, 27.1.7] that the d -th symmetric power S d V (i.e., the simple module ofhighest weight being d times the first fundamental weight) is a based submodule of V ⊗ d inthe sense of [L93, Chap. 27], and hence S d V ⊗ · · · ⊗ S d s V is also a based submodule of V ⊗ d ,where the positive integers d i satisfy d + . . . + d s = d . The following is now a consequence(and also a generalization) of Corollary 3.7. Corollary 3.8. The action of ˙ U ( sl n ) on S d V ⊗ · · · ⊗ S d s V with respect to the correspondingcanonical bases is positive. OSITIVITY VS NEGATIVITY OF CANONICAL BASES 13 A positive basis for ˙ U ( gl n ) . Note that the BLM basis of ˙ U ( gl n ) restricts to a ba-sis of ˙ U ( gl n ) h d i , which does not have positive structure constants in general by Propo-sition 2.6. However, in light of the positivity in Proposition 3.1, one can transport thecanonical basis on ˙ U ( sl n ) h d i to ˙ U ( gl n ) h d i via the isomorphism ℘ d in (3.3), which has pos-itive structure constants. Let us denote the resulting positive basis (or can ⊕ nical basis ) on˙ U ( gl n ) = ⊕ d ∈ Z ˙ U ( gl n ) h d i by B pos ( gl n ). By definition, the basis B pos ( gl n ) is invariant underthe shift maps ξ p for p ∈ Z . Summarizing we have the following. Proposition 3.9. There exists a positive basis B pos ( gl n ) for A ˙ U ( gl n ) (and also for ˙ U ( gl n ) ),which is induced from the canonical basis for A ˙ U ( sl n ) . Recall a 2-category ˙ U ( gl n ) which categorifies ˙ U ( gl n ) in [MSV13] is obtained by simply rela-beling the objects for the Khovanov-Lauda 2-category which categorifies ˙ U ( sl n ) in [KhL10].We expect that the projective indecomposable 1-morphisms in ˙ U ( gl n ) categorify the positivebasis B pos ( gl n ) (instead of the BLM basis which has no positivity).4. Modified quantum coideal algebras ˙ U ( gl n ) and ˙ U ( sl n ) , for n odd In this section and next section, we fix 2 odd positive integers n, D such that n = 2 r + 1 , D = 2 d + 1 . We will almost exclusively use the notation n and d (instead of r and D ). We study thecanonical bases for the modified quantum coideal algebras ˙ U ( gl n ) and ˙ U ( sl n ) as well asthe Schur algebras S ( n, d ). We will again use the notation { A } , [ A ] , { A } d etc for the basesof these algebras, as these sections are independent from the earlier ones to a large extent.When we occasionally need to refer to similar bases in type A from earlier sections, we shalladd a superscript a .In this section, we show that the stably canonical basis constructed in [BKLW] for themodified quantum coideal algebra ˙ U ( gl n ) does not have positive structure constants. Wealso formulate some basic connections between ˙ U ( gl n ) and ˙ U ( sl n ).4.1. Schur algebras and quantum coideal algebra. We first recall some basics from[BKLW].Let F q be a finite field of odd order q . Let A S ( n, d ) (denoted by S in [BKLW]) be the Schur algebra over A , which specializes at v = √ q to the convolution algebra of pairs of n -step partial isotropic flags in F d +1 q (with respect to some fixed non-degenerate symmetricbilinear form). The algebra A S ( n, d ) admits a bar involution, a standard basis [ A ] d , and acanonical (= IC) basis { A } d parameterized by(4.1) Ξ d = n A = ( a ij ) ∈ Θ d +1 (cid:12)(cid:12) a ij = a n +1 − i,n +1 − j , ∀ i, j ∈ [1 , n ] o . Set Ξ := ∪ d ≥ Ξ d . The multiplication formulas of the A -algebras A S ( n, d ) exhibits some remarkable stabilityas d varies, which leads to a “limit” A -algebra K . The bar involution on A S ( n, d ) inducesa bar involution on K [BKLW, § K has a standard basis [ A ] and a stably canonical basis { A } , parameterized by˜Ξ = n A = ( a ij ) ∈ Mat n × n ( Z ) | a ij ≥ i = j ) ,a r +1 ,r +1 ∈ Z + 1 , a ij = a n +1 − i,n +1 − j ( ∀ i, j ) o . (4.2)Recall (cf. [BW13, BKLW] and the references therein) there is a quantum coideal algebra U ( gl n ) which can be embedded in U ( gl n ), and ( U ( gl n ) , U ( gl n )) form a quantum symmetricpair in the sense of Letzter. For our purpose here, its modified version ˙ U ( gl n ) is more directlyrelevant; we recall its presentation below from [BKLW, § Z n = n µ ∈ Z n | µ i = µ n +1 − i ( ∀ i ) and µ ( n +1) / is odd o . Let E θij be the n × n matrix whose ( k, l )-entry is equal to δ k,i δ l,j + δ k,n +1 − i δ l,n +1 − j . Given λ ∈ Z n , we introduce the short-hand notation λ ± α i whose j th entry is equal to λ j ∓ ( δ i,j + δ n +1 − i,j ) ± ( δ i +1 ,j + δ n − i,j ). Recall n = 2 r + 1. The algebra ˙ U ( gl n ) is the Q ( v )-algebragenerated by 1 λ , e i λ , 1 λ e i , f i λ and 1 λ f i , for i = 1 , . . . , r and λ ∈ Z n , subject to thefollowing relations, for i, j = 1 , . . . , ( n − / λ, λ ′ ∈ Z n : x λ λ ′ x ′ = δ λ,λ ′ x λ x ′ , for x, x ′ ∈ { , e i , e j , f i , f j } ,e i λ = 1 λ − α i e i ,f i λ = 1 λ + α i f i ,e i λ f j = f j λ − α i − α j e i , if i = j,e i λ f i = f i λ − α i e i + v λi +1 − λi − v λi − λi +1 v − v − λ − α i , if i = n − , ( e i e j + e j e i )1 λ = ( v + v − ) e i e j e i λ , if | i − j | = 1 , ( f i f j + f j f i )1 λ = ( v + v − ) f i f j f i λ , if | i − j | = 1 ,e i e j λ = e j e i λ , if | i − j | > ,f i f j λ = f j f i λ , if | i − j | > , ( f r e r − ( v + v − ) f r e r f r + e r f r )1 λ = − ( v + v − ) (cid:16) v λ r +1 − λ r − + v λ r − λ r +1 +2 (cid:17) f r λ , ( e r f r − ( v + v − ) e r f r e r + f r e r )1 λ = − ( v + v − ) (cid:16) v λ r +1 − λ r +1 + v λ r − λ r +1 − (cid:17) e r λ . It was shown in [BKLW, § A -algebra isomorphism K ∼ = A ˙ U ( gl n ),which matches the Chevalley generators. we shall always make such an identification K ≡ A ˙ U ( gl n ) and use only A ˙ U ( gl n ) in the remainder of the paper.Given m ∈ Z with 0 ≤ m ≤ n , let J m be an m × m matrix whose ( i, j )-th entry is δ i,m +1 − j . Recalling the definition of ˜Θ depends on n from Section 2.1, we shall write ˜Θ n for˜Θ in this paragraph and allow n vary, and so in particular ˜Θ m makes sense. To a matrix A ∈ ˜Θ m and k ∈ Z , we define a matrix τ km,n ( A ) = A kI + ε 00 0 J m AJ m where ε is the ( n − m ) × ( n − m ) matrix whose only nonzero entry is the very central one,which equals 1. Thus, we have an embedding τ km,n : ˜Θ m −→ ˜Ξ , A τ km,n ( A ) . OSITIVITY VS NEGATIVITY OF CANONICAL BASES 15 By comparing the multiplication formulas [BLM90, 4.6] in A ˙ U ( gl m ) and those in A ˙ U ( gl n )[BKLW, (4.5)-(4.7)], we have an algebra embedding, also denoted by τ km,n , τ km,n : A ˙ U ( gl m ) −→ A ˙ U ( gl n ) , a [ A ] [ τ km,n ( A )] . (4.3)(We recall here our convention of using the superscript a to denote the corresponding basisin the type A setting from earlier sections.) Note that the homomorphism τ km,n commuteswith the bar involutions on A ˙ U ( gl m ) and A ˙ U ( gl n ). The following lemma is immediate fromthe definitions. Lemma 4.1. Suppose that ≤ m ≤ ( n − / and k ∈ Z . Then τ km,n ( a { A } ) = { τ km,n ( A ) } ,for all A ∈ ˜Θ m . We denote S ( n, d ) = Q ( v ) ⊗ A A S ( n, d ) , ˙ U ( gl n ) = Q ( v ) ⊗ A A ˙ U ( gl n ) . The quantum coideal algebra U ( sl n ) can be embedded into (and hence identified with asubalgebra of) U ( sl n ); cf. [BW13]. We define an equivalence relation ∼ on Z n : µ ∼ µ ′ if µ − µ ′ = m P ni =1 ǫ i for some m ∈ Z . Let ¯ µ denote the equivalence class of µ . Put ∧ Z n = Z n / ∼ . We define the Q ( v )-algebra ˙ U ( sl n ) formally in the same way as ˙ U ( gl n ) above except nowthat the weights λ, λ ′ run over ∧ Z n (instead of Z n ). There exists a bar involution on ˙ U ( sl n )(as well as on ˙ U ( gl n )) which fixes all the generators. The A -form A ˙ U ( sl n ) of the Q ( v )-algebra ˙ U ( sl n ) (as well as the A -form A ˙ U ( gl n ) of ˙ U ( gl n )) is generated by the dividedpowers e ( a ) i λ , f ( a ) i λ for all admissible i, a, λ .For later use we define an equivalence relation ∼ on ˜Ξ: A ∼ A ′ if and only if A − A ′ = mI, for some m ∈ Z . We set(4.4) b Ξ = ˜Ξ / ∼ . Negativity of stably canonical basis for ˙ U ( gl n ) . For a, b ∈ Z , let A = a b 00 1 a , B = a b 10 0 a , C = a − b 10 1 a − , D = a b + 2 00 0 a . The following example arises from discussions with Huanchen Bao. Proposition 4.2. The structure constants for the stably canonical basis of ˙ U ( gl n ) are notalways positive, for n ≥ . More explicitly, for n = 3 and for a, b ∈ Z with a < b ≤ − , thefollowing identity holds in ˙ U ( gl ) : { B } ∗ { A } = { C } + ( v b + a + v b − a )[ b + 1] { D } where [ b + 1] ∈ Z ≤ [ v, v − ] .Proof. It suffices to check the identity for n = 3, since the general case for n ≥ B ] ∗ [ A ] = [ C ] + v − a v b [ b + 1][ D ] . (4.5) Observe that { D } = [ D ] , { A } = [ A ] , { B } = [ B ]since D is diagonal, [ A ] and [ B ] are the Chevalley generators of ˙ U ( gl ). Also note that v b [ b + 1] is a bar-invariant quantum integer. Applying the bar involution to (4.5) and com-paring with (4.5) again, we have(4.6) [ C ] − [ C ] = ( v − a − v a ) v b [ b + 1][ D ] . By assumption that a < b ≤ − 2, we have v a + b [ b + 1] ∈ v − Z < [ v − ], and hence from (4.6)we obtain that { C } = [ C ] − v a + b [ b + 1][ D ] . Now the equation (4.5) can be rewritten as { B } ∗ { A } = { C } + ( v a + v − a ) v b [ b + 1][ D ] . It is clear that v b [ b + 1] = − ( v − b + v − b − + . . . + v b +2 + v b ) ∈ Z ≤ [ v, v − ] for b ≤ − 2. Thisfinishes the proof for n = 3. (cid:3) Relating ˙ U ( gl n ) to ˙ U ( sl n ) . This subsection, in which we are making a transitionfrom ˙ U ( gl n ) to ˙ U ( sl n ), is a preparation for the next section.Recall that there is a Schur ( S ( n, d ) , H S d )-duality on V ⊗ d , where V is an n -dimensionalvector space over Q ( v ). It is shown [G97, BW13] (see also [BKLW]) that there is a Schur-type ( S ( n, d ) , H B d )-duality on V ⊗ d where H B d is the Iwahori-Hecke algebra associated tothe hyperoctahedral group B d . In particular we have algebra homomorphisms S ( n, d ) ∼ = −→ End H Sd ( V ⊗ d ) , S ( n, d ) ∼ = −→ End H Bd ( V ⊗ d ) . Recall the sign homomorphism(4.7) χ n : S ( n, n ) −→ Q ( v )from the proof of Proposition 3.3 (cf. [L00, 1.8]). We have a natural inclusion of algebras H B d × H S n ⊆ H B d + n . The transfer map φ d + n,d : S ( n, d + n ) −→ S ( n, d )is defined as the composition of the homomorphisms S ( n, d + n ) ∼ = −→ End H Bd + n ( V ⊗ ( d + n ) ) ∆ −→ End H Bd × H Sn ( V ⊗ ( d + n ) ) ∼ = −→ End H Bd ( V ⊗ d ) ⊗ End H Sn ( V ⊗ n ) ⊗ χ n −→ End H Bd ( V ⊗ d ) ∼ = −→ S ( n, d ) . (4.8)This transfer map will be studied in depth from a geometric viewpoint in [FL15], where theproof of the following lemma can be found. Lemma 4.3. We have φ d + n,d ([ A ] d + n ) = ( [ A − I ] d , if A − I ∈ Ξ d , , otherwise.for all A ∈ Ξ d + n such that one of the following matrices is diagonal: A , A − aE θi +1 ,i or A − aE θi,i +1 for some a ∈ N and ≤ i ≤ ( n − / . OSITIVITY VS NEGATIVITY OF CANONICAL BASES 17 Similar to the decomposition (2.1) for ˙ U ( gl n ), we can decompose ˙ U ( gl n ) as a direct sumof subalgebras ˙ U ( gl n ) = M d ∈ Z ˙ U ( gl n ) h d i , where ˙ U ( gl n ) h d i is spanned by elements of the form 1 λ u µ with | µ | = | λ | = 2 d + 1 and u ∈ ˙ U ( gl n ) . Also similar to the decomposition (3.2) for ˙ U ( sl n ), we can decompose ˙ U ( sl n )as a direct sum of n subalgebras ˙ U ( sl n ) = M ¯ d ∈ Z /n Z ˙ U ( sl n ) h ¯ d i , where ˙ U ( sl n ) h ¯ d i is spanned by 1 µ ˙ U ( sl n )1 λ , where | µ | ≡ | λ | ≡ d + 1 mod 2 n . Denote by π ¯ d : ˙ U ( sl n ) → ˙ U ( sl n ) h ¯ d i the natural projection. There exists a natural algebra isomorphismsimilar to (3.3)(4.9) ℘ d, : ˙ U ( gl n ) h d i ∼ = ˙ U ( sl n ) h d i ( ∀ d ∈ Z ) , which induces a homomorphism ℘ : ˙ U ( gl n ) → ˙ U ( sl n ). In the same way as for ˙ U ( gl n )defined in (2.2), for each p ∈ Z we define a shift map ξ p : ˙ U ( gl n ) −→ ˙ U ( gl n ) , ξ p ([ A ]) = [ A + pI ] , (4.10)where either A , A − E θh,h +1 or A − E θh +1 ,h for some 1 ≤ h ≤ n − ℘ ◦ ξ p = ℘ , for all p ∈ Z . Recall a homomorphism Φ d : ˙ U ( gl n ) → S ( n, d ) was defined in [BKLW, § φ d therein) which sends [ A ] to [ A ] d for A ∈ Ξ d and to zero otherwise. We define φ d : ˙ U ( sl n ) −→ S ( n, d )to be the composition(4.12) ˙ U ( sl n ) π ¯ d −→ ˙ U ( sl n ) h ¯ d i ℘ − d, −→ ˙ U ( gl n ) h d i Φ d −→ S ( n, d ) . We introduce another homomorphismΨ d : ˙ U ( gl n ) −→ S ( n, d )to be the composition of the following homomorphisms˙ U ( gl n ) ℘ −→ ˙ U ( sl n ) φ d −→ S ( n, d ) . Note that Ψ d = Φ d , but Ψ d coincides with Φ d when restricted to ˙ U ( gl n ) h d i . Proposition 4.4. We have the following commutative diagram: ˙ U ( gl n ) ℘ (cid:15) (cid:15) Ψ d + n | | ②②②②②②②②②②②②②②②②②②②②②②② Ψ d ! ! ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ ˙ U ( sl n ) φ d + n u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ φ d ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ S ( n, d + n ) φ d + n,d / / S ( n, d )(4.13) Proof. The commutativity of the left upper triangle and the right upper triangle is clearfrom definition. The commutativity of the bottom triangle follows from a description of thehomomorphisms φ d and φ d + n,d in terms of matching the generators by Lemma 4.3. (cid:3) Canonical basis for modified quantum coideal algebra ˙ U ( sl n ) , for n odd In this section we continue (as in Section 4) to let n = 2 r + 1 and D = 2 d + 1 be oddpositive integers.We establish some asymptotical behavior for the canonical bases of Schur algebras underthe transfer map. This is used to define the canonical basis for ˙ U ( sl n ) and to show thatstructure constants of the canonical basis of ˙ U ( sl n ) are positive. We further show that thetransfer map on the Schur algebras sends every canonical basis element to a positive sumof canonical basis elements or zero, and provide some corollaries.5.1. Asymptotic identification of canonical bases for S ( n, d ) . Recall a bilinear form h· , ·i d on S ( n, d ) is defined in [BKLW, § · , · ) D therein with D = 2 d + 1).The same argument as for [M12, Proposition 4.3] shows that(5.1) h x, y i := lim p →∞ n − X d =0 (cid:10) φ d + pn ( x ) , φ d + pn ( y ) (cid:11) d + pn , for x, y ∈ ˙ U ( sl n ) , exists as an element in Q ( v ). Thus we have constructed a bilinear form h− , −i on ˙ U ( sl n ).Recall there is a partial order (cid:22) on ˜Ξ [BKLW, (3.22)] by declaring A (cid:22) B if and only if P r ≤ i ; s ≥ j a rs ≤ P r ≤ i ; s ≥ j b rs for all i < j. For an n × n matrix A = ( a ij ), letro( A ) = (cid:16) X j a j , X j a j , . . . , X j a nj (cid:17) , co( A ) = (cid:16) X i a i , X i a i , . . . , X i a in (cid:17) . There is a partial order ⊑ on ˜Ξ [BKLW, (3.24)], which refines (cid:22) , so that A ′ ⊑ A if and onlyif A ′ (cid:22) A , ro( A ′ ) = ro( A ) and co( A ′ ) = co( A ). The following lemma is preparatory. Lemma 5.1. Fix A = ( a ij ) ∈ ˜Ξ . Suppose that p is an even integer such that a ll + p ≥ P i = j a ij for all ≤ l ≤ n . If B ∈ ˜Ξ satisfies B ⊑ p A , then B ∈ Ξ | p A | , i.e., b ii ≥ for all ≤ i ≤ n .Proof. We prove by contradiction. Suppose that b i ,i < i . We have X j = i b i j > ro( B ) i = ro( p A ) i ≥ a i i + p ≥ X i = j a ij . OSITIVITY VS NEGATIVITY OF CANONICAL BASES 19 This implies that X r ≤ i ,s ≥ i +1 b rs + X r ≥ i ,s ≤ i − b rs ≥ X j = i b i j > X i = j a ij ≥ X r ≤ i ,s ≥ i +1 a rs + X r ≥ i ,s ≤ i − a rs , which contradicts with the condition B ⊑ p A . (cid:3) Proposition 5.2. Given A ∈ ˜Ξ with | A | = 2 d + 1 , we have, for even integers p ≫ , φ d,d − n ( { p A } d ) = { ( p − A } d − n , where we denote d = d + pn/ so that | p A | = 2 d + 1 . Proof. The proof is essentially adapted from that of [M12, Proposition 7.8] with minor mod-ifications. Let us go over it for the sake of completeness.Recall the monomial basis { d M A | A ∈ Ξ d } of S ( n, d ) from [BKLW, (3.25)], (which is denotedby m A therein). By Lemma 4.3 we have φ d,d − n ( d M A ) = d − n M A − I , ∀ d. (It is understood that d − n M A − I = 0 if A − I Ξ d − n .) The proposition is equivalent to thefollowing. Claim ( ⋆ ). Let A ∈ ˜Ξ . For all even integer p ≫ , we have { p A } d = d M p A + X A ′ ≺ A c A ′ ,A,p d M p A ′ , where c A ′ ,A,p ∈ A is independent of p ≫ . Recall [BKLW] that the basis { d M p A } satisfies d M p A = d M p A , d M p A ∈ A S ( n, d ), and(5.2) d M p A = { p A } d + X B ≺ A w p A, p B { p B } d , for some w p A, p B ∈ A . We shall argue similarly as for a claim in the proof of [M12, Proposition 7.8], with D b A used in loc. cit. replaced by d M p A ; that is, we shall prove Claim ( ⋆ ) by induction on A withrespect to the partial order (cid:22) . When A is minimal, it follows by (5.2) that d M p A = { p A } d forall p , and hence Claim ( ⋆ ) holds.Now assume that Claim ( ⋆ ) holds for all B such that B ≺ A . Set I d = (cid:8) B ∈ ˜Ξ (cid:12)(cid:12) B (cid:22) A, p B ∈ Ξ d , ro( B ) = ro( A ) , co( B ) = co( A ) (cid:9) . Then for p ≫ 0, we have by Lemma 5.1 that • I d = { B ∈ ˜Ξ | B (cid:22) A, ro( B ) = ro( A ) , co( B ) = co( A ) } ; • I d is a finite set, and it is independent of p ≫ d = d + pn/ p ).For u ∈ A = Z [ v, v − ], let deg( u ) be its degree. For x ∈ Span A {{ p B } d | B ∈ I d } , we set n ( x ) = max (cid:8) deg h x, { p B } d i d (cid:12)(cid:12) B ∈ I d , B = A (cid:9) , and n p = n ( d M p A ) . Suppose that n p ≥ 0. We set J d = (cid:8) B ∈ I d (cid:12)(cid:12) deg h d M p A , { p B } d i d = n p (cid:9) . Then we can write, for each B ∈ I d , (cid:10) d M p A , { p B } d (cid:11) d = X i ≤ n p c B,p,i v i ∈ Z [ v, v − ] , where c B,p,i ∈ Z ( ∀ i ) , and c B,p,n p ( = 0 , if B ∈ J d , = 0 , if B ∈ I d \J d . (5.3)We define a new bar-invariant element in A S ( n, d ): d M ′ p A = ( d M p A − P B ∈J d c B,p,n p ( v n p + v − n p ) { p B } d , if n p > , d M p A − P B ∈J d c B,p,n p { p B } d , if n p = 0 . We now show that n ( d M ′ p A ) < n p = n ( d M p A ). We give the details for n p > 0, while the casefor n p = 0 is entirely similar. By the almost orthonormality of the canonical basis of S ( n, d )[BKLW], we have (cid:10) { p B } d , { p B ′ } d (cid:11) d ∈ δ B,B ′ + v − Z [ v − ]. For B ∈ I d , we have by (5.3) that (cid:10) d M ′ p A , { p B } d (cid:11) d = (cid:10) d M p A , { p B } d (cid:11) d − X B ′ ∈J d c B ′ ,p,n p ( v n p + v − n p ) (cid:10) { p B } d , { p B ′ } d (cid:11) d ≡ X i ≤ n p − c B,p,i v i − X B = B ′ ∈J d c B ′ ,p,n p v n p (cid:10) { p B } d , { p B ′ } d (cid:11) d mod v − Z [ v − ] , which implies that n ( d M ′ p A ) < n p .By repeating the above procedure with d M ′ p A in place of d M p A , we produce a bar-invariantelement d M ′′ p A in A S ( n, d ) with degree n ( d M ′′ p A ) < n ( d M ′ p A ), and then repeat again and so on.So under the assumption that n p ≥ 0, after finitely many steps we obtain a bar-invariantelement in A S ( n, d ), denoted by b p A , with n ( b p A ) < n p = n ( d M p A ) < 0, then we simply set b p A = d M p A .We now show that b p A = { p A } d . By the above construction and (5.2), we have b p A = { p A } d + X B ∈I d f B { p B } d , for some f B ∈ A and f B = f B . If f B = 0 for some B , then n ( b p A ) ≥ 0, which is acontradiction. Hence we have b p A = { p A } d .In the finite process above of constructing { p A } d (in the form of b p A ) from the monomialbasis, we only need the first n p coefficients of h d M p A , { p B } d i d as well as of h{ p B ′ } d , { p B } d i d for B ∈ I d , B ′ ∈ J d . Recall that the monomial basis { M A | A ∈ ˜Ξ } of K from [BKLW, 4.8]satisfies that φ d ( M A ) = d M p A if p A ∈ Ξ d . So by the inductive assumption that any element B ≺ A satisfies Claim ( ⋆ ) and the convergence of the bilinear form h· , ·i d (with d = d + pn/ Q (( v − )) as p 7→ ∞ , we conclude that I d , n p and c B,p,i (0 ≤ i ≤ n p ) are all independent of p ≫ 0. Now Claim ( ⋆ ) follows by the construction of { p A } d as b p A in terms of the monomialbasis above. (cid:3) Proposition 5.3. Given A ∈ ˜Ξ , we have ξ − ( { p A } ) = { ( p − A } , ℘ ( { p A } ) = ℘ ( { ( p − A } ) for all even integers p ≫ , where ξ − is defined in (4.10). OSITIVITY VS NEGATIVITY OF CANONICAL BASES 21 Proof. Denote | A | = 2 d + 1, and d = d + pn/ . We have the following commutative diagram˙ U ( gl n ) ξ − −−−→ ˙ U ( gl n ) Φ d y Φ d − n y S ( n, d ) φ d,d − n −−−−→ S ( n, d − n )(5.4)i.e., Φ d − n ◦ ξ − = φ d,d − n ◦ Φ d . By [BKLW, Appendix A, Theorem A.21], we have(5.5) Φ d ( { p A } ) = { p A } d , Φ d − n ( { ( p − A } ) = { ( p − A } d − n , ∀ p ≫ . Moreover, by [BKLW, (4.8)], we have(5.6) ξ − ( { p A } ) = { ( p − A } + X B ∈ Ξ d − n f B { B } , (for f B ∈ A ) , where the summation can be taken over B ∈ Ξ d − n is ensured by Lemma 5.1.Using Proposition 5.2, (5.5), (5.4), and (5.6) one by one, we conclude that { ( p − A } d − n = φ d,d − n ◦ Φ d ( { p A } )= Φ d − n ◦ ξ − ( { p A } ) = { ( p − A } d − n + X B ∈ Ξ d − n B ❁ ( p − A f B { B } d − n . Hence all f B must be zero, and the first identity in the proposition follows from (5.6). Thesecond identity is immediate from the first one and (4.11). (cid:3) Canonical basis for ˙ U ( sl n ) . By Proposition 5.3, for b A ∈ b Ξ (recall b Ξ from (4.4)), theelement b b A := ℘ ( { p A } ) , for p ≫ p and thus a well-defined element in ˙ U ( sl n ). It follows by definition that ℘ : ˙ U ( gl n ) → ˙ U ( sl n ) preserves the A -forms, so we have b b A ∈ A ˙ U ( sl n ). Proposition 5.4. For A ∈ ˜Ξ with | A | = 2 d + 1 , let d = d + pn/ . Then φ d ( b b A ) = { p A } d for even integers p ≫ .Proof. We have, for p ≫ φ d ( b b A ) = φ d ( ℘ ( { p A } )) = Ψ d ( { p A } ) = Φ d ( { p A } ) = { p A } d , where the first equality follows by definition, the second one is due to (4.13), the third onefollows by definition (4.12), and the last one follows from [BKLW, Theorem 6.10]. Theproposition is proved. (cid:3) Theorem 5.5. The set ˙ B ( sl n ) = { b b A | b A ∈ b Ξ } forms a basis of ˙ U ( sl n ) , and it also formsan A -basis for A ˙ U ( sl n ) .Proof. Observe that ξ p ( { A } ) = { A + pI } + lower terms . Hence it follows by the surjectivityof ℘ that ˙ B ( sl n ) is a spanning set for the A -module A ˙ U ( sl n ). To show that ˙ B ( sl n ) islinearly independent, it suffices to check that ˙ B ( sl n ) ∩ ˙ U ( sl n ) h ¯ d i is linearly independent foreach ¯ d ∈ Z /n Z . This is then reduced to the Schur algebra level by Proposition 5.4, which is clear. Hence ˙ B ( sl n ) = { b b A | b A ∈ b Ξ } is an A -basis of A ˙ U ( sl n ), and thus it is also a basis of˙ U ( sl n ). (cid:3) Positivity of the canonical basis ˙ B ( sl n ) . The basis ˙ B ( sl n ) is called the canonicalbasis (or -canonical basis ) of ˙ U ( sl n ), as we shall show that the canonical basis ˙ B ( sl n )admits several remarkable properties such as positivity and almost orthonormality just likeLusztig’s canonical basis for ˙ U ( sl n ) (see Proposition 3.1 and [L93]).Given b A, b B ∈ b Ξ, we write b b A ∗ b b B = X b C ∈ b Ξ P b C b A, b B b b C , where P b C b A, b B ∈ Z [ v, v − ] is zero for all but finitely many b C . Theorem 5.6 (Positivity) . We have P b C b A, b B ∈ N [ v, v − ] , for any b A, b B, b C ∈ b Ξ .Proof. Let us write b b A ∗ b b B = P b C ∈ Ω P b C b A, b B b b C , where Ω is the finite set which consists of b C ∈ b Ξsuch that P b C b A, b B = 0. Let us pick representatives A, B, C ∈ ˜Ξ such that | A | = | B | = | C | =2 d + 1 for all b C ∈ Ω.By Proposition 5.4, we can find some large p (and recall d = d + pn/ 2) such that p A, p B, p C ∈ Ξ and φ d ( b b A ) = { p A } d , φ d ( b b B ) = { p B } d , φ d ( b b C ) = { p C } d , for all C with b C ∈ Ω. So we have the following multiplication of canonical basis in S ( n, d ): { p A } d ∗ { p B } d = X b C ∈ Ω P b C b A, b B { p C } d . Thanks to the intersection cohomology construction of the canonical basis for S ( n, d ) [BKLW],the structure constants P b C b A, b B lie in N [ v, v − ]. This proves the theorem. (cid:3) Proposition 5.7. The bilinear form h· , ·i on ˙ U ( sl n ) is non-degenerate. Moreover, thealmost orthonormality for the canonical basis holds: h b b A , b b B i ∈ δ b A, b B + v − Z [[ v − ]] .Proof. This almost orthonormality follows by an argument entirely similar to [M12, Theo-rem 8.1], and it implies the non-degeneracy of the bilinear form. (cid:3) We have the following positivity for the canonical bases with respect to the bilinear form. Theorem 5.8. We have h b b A , b b B i = δ b A, b B + v − N [[ v − ]] , for any b A, b B ∈ b Ξ .Proof. The proof follows very closely McGerty’s geometric argument [M12, Proposition 6.5,Theorem 8.1], with [M12, Corollary 3.3] replaced by [BKLW, Corollary 3.15]. We only sketchthe proof with an emphasis on the difference and refer to loc. cit. for further details.By the definition of h· , ·i , it is reduced to show that h{ A } d , { B } d i d ∈ δ A,B + v − N [ v − ] forall A, B ∈ Ξ d where h· , ·i d is the bilinear form on S ( n, d ). The positivity of the form h· , ·i d in the theorem will follow by its identification with another geometrically defined bilinearform h· , ·i g,d on S ( n, d ) which manifests the positivity. The latter is defined exactly the sameas [M12, (6-1)] with the flag variety F a therein replaced by the n -step isotropic flag variety OSITIVITY VS NEGATIVITY OF CANONICAL BASES 23 of a (2 d + 1)-dimensional complex vector space equipped with a non-degenerate symmetricbilinear form.Now arguing similar to [M12, Lemma 6.3], we have, for all A minimal with respect to thepartial order ≤ , h{ A } d ∗ { B } d , { C } d i g,d = v d A − d At h{ B } d , { A t } d ∗ { C } d i g,d , where A t is the transpose of A . This implies the analog of [M12, Lemma 6.4], which givesthe formulas for the adjoints of the Chevalley generators of S ( n, d ) for the bilinear form h· , ·i g,d , and we observe that they coincide with the ones for h· , ·i d given in [BKLW, Corollary3.15]. Hence, the identification of the forms h· , ·i d and h· , ·i g,d is reduced to show that h{ A } d , { D λ } d i d = h{ A } d , { D λ } d i g,d , ∀ A, λ where D λ is the diagonal matrix with diagonal λ . Indeed, if we write { A } d = P A ′ ≤ A P A,A ′ [ A ′ ] d for some P A,A ′ ∈ Z [ v − ], then both sides of the above equation are equal to P A,D λ ifro( A ) = co( A ) = λ , or zero otherwise. The theorem follows. (cid:3) Furthermore, we have the following characterization of the signed canonical basis. Proposition 5.9. The signed canonical basis − ˙ B ( sl n ) ∪ ˙ B ( sl n ) is characterized by thefollowing three properties: (i) b = b , (ii) b ∈ A ˙ U ( sl n ) , and (iii) h b, b ′ i ∈ δ b,b ′ + v − Z [[ v − ]] .Proof. It follows by definition and Proposition 5.7 that − ˙ B ( sl n ) ∪ ˙ B ( sl n ) satisfies the threeproperties above. The characterization claim is then proved in the same way as [L93, 14.2.3]for the usual canonical bases. (cid:3) Positivity of transfer map φ d + n,d . We have the following positivity on the transfermap φ d + n,d , generalizing Proposition 3.3 on the positivity of the transfer map φ d + n,d . Theorem 5.10. The transfer map φ d + n,d : S ( n, d + n ) → S ( n, d ) sends each canonical basiselement to a sum of canonical basis elements with (bar invariant) coefficients in N [ v, v − ] .Proof. The strategy of the proof is identical to the one for Proposition 3.3, which is reducedto the positivity of ∆ defined in (4.8) with respect to the canonical bases and the positivityof χ which was already established in (3.8). The proof of the positivity of ∆ is similar tothat of ∆ in the proof of Proposition 3.3 (the details are provided in [FL15] together withother applications in a geometric setting). (cid:3) Proposition 5.11. The map φ d : ˙ U ( sl n ) → S ( n, d ) sends each canonical basis element toa sum of canonical basis elements with (bar invariant) coefficients in N [ v, v − ] .Proof. This follows by applying (4.13), Proposition 5.4 and Theorem 5.10. The detail iscompletely analogous to the proof of Proposition 3.4 and hence skipped. (cid:3) Remark 5.12. Theorem 5.10 provides a strong evidence for a possible functor realization ofthe transfer map φ d + n,d (cf. [M12, Remark 7.10]). In light of [L99, SV00], it is interesting tosee if φ d + n,d (and hence φ d ) sends each canonical basis element to a canonical basis elementor zero, improving Theorem 5.10 and Proposition 5.11; compare with Remark 3.5. Recall there is a Schur-type ( S ( n, d ) , H B d )-duality on V ⊗ d [G97, BW13], where V is n -dimensional, and this duality can be completely realized geometrically [BKLW]. Denote by B ( n d ) the -canonical basis of V ⊗ d constructed in [BW13]. These canonical bases on V ⊗ d as well as on S ( n, d ) are realized in [BKLW] as simple perverse sheaves, and the action of S ( n, d ) on V ⊗ d is realized in terms of a convolution product. Hence we have the followingpositivity. Proposition 5.13. The action of S ( n, d ) on V ⊗ d with respect to the corresponding -canonical bases is positive in the following sense: for any canonical basis element a of S ( n, d ) and any b ∈ B ( n d ) , we have a ∗ b = X b ′ ∈ B ( n d ) D b ′ a,b b ′ , where D b ′ a,b ∈ N [ v, v − ] . We obtain a natural action of ˙ U ( sl n ) on V ⊗ d by composing the action of S ( n, d ) on V ⊗ d with the map φ d : ˙ U ( sl n ) → S ( n, d ). As a corollary of Propositions 5.11 and 5.13 we havethe following positivity (which is a special case of a conjectural positivity property of thecanonical basis for general tensor product modules [BW13]). Corollary 5.14. The action of ˙ U ( sl n ) on V ⊗ d with respect to the corresponding -canonicalbases is positive. Compatibility of canonical bases ˙ B ( sl m ) and ˙ B ( sl n ) . Given integers k, m with0 ≤ m ≤ n , we recall τ km,n from (4.3). Fix an m -tuple of integers k = ( k , k , . . . , k m − ).We define an imbedding τ k d m,n : ˙ U ( sl m ) h d i → ˙ U ( sl n ) h d + k d ( n − m ) i , for 0 ≤ d < m , to bethe composition(5.7) ˙ U ( sl m ) h d i ℘ − d −→ ˙ U ( gl m ) h d i τ kdm,n −→ ˙ U ( gl n ) h d + k d ( n − m ) i ℘ −→ ˙ U ( sl n ) h d + k d ( n − m ) i . These τ k d m,n for all d can be combined into a homomorphism τ k m,n : ˙ U ( sl m ) → ˙ U ( sl n ). Werecall Θ m from (3.1), which is understood in this subsection to consist of m × m matrices. Proposition 5.15. Retaining the notations above, we have τ k m,n (cid:0) ˙ B ( sl m ) (cid:1) ⊆ ˙ B ( sl n ) . Moreprecisely, if b A ∈ ˙ B ( sl m ) for A ∈ Θ m , then τ k m,n ( b A ) = b c A ′ , where A ′ = τ k d m,n ( A ) if | A | = d .Proof. We have the following commutative diagram:˙ U ( gl m ) h d i τ kdm,n −−−→ ˙ U ( gl n ) h d + k d ( n − m ) i ξ l y ξ l y ˙ U ( gl m ) h d + 2 lm i τ kd + lm,n −−−→ ˙ U ( gl n ) h d + k d ( n − m ) + ln i Let A ∈ Θ m . Pick the preimage (an m × m matrix) A of A with 0 ≤ | A | < m , and set d = | A | .Recall from (3.4) and (4.11) that ℘ ◦ ξ l = ℘ and ℘ ◦ ξ l = ℘ , for l ∈ Z . It follows fromthese identities, (5.7), and the above commutative diagram that τ k d m,n = ℘ ◦ τ k d + lm,n ◦ ℘ − d +2 lm .Hence applying [M12, Proposition 7.8], Lemma 4.1, and Proposition 5.3 in a row give us (for l ≫ τ k d m,n ( b A ) = ℘ ◦ τ k d + lm,n ◦ ℘ − d +2 lm ( b A ) = ℘ ◦ τ k d + lm,n ( a { l A } ) = ℘ ( { τ k d + lm,n ( l A ) } ) = b c A ′ , OSITIVITY VS NEGATIVITY OF CANONICAL BASES 25 where the last identity uses the fact that A ′ = τ k d m,n ( A ) and τ k d + lm,n ( l A ) have the same imagein b Ξ. The proposition is proved. (cid:3) A positive basis for ˙ U ( gl n ) . Recall that the stably canonical basis of ˙ U ( gl n ) (andhence of ˙ U ( gl n ) h d i for d ∈ Z ) does not have positive structure constants in general byProposition 4.2. However, one can transport the canonical basis on ˙ U ( sl n ) h d i to ˙ U ( gl n ) h d i via the isomorphism ℘ d, in (4.9), which has positive structure constants by Theorem 5.6. Letus denote the resulting positive basis (or can ⊕ nical basis ) on ˙ U ( gl n ) = ⊕ d ∈ Z ˙ U ( gl n ) h d i by B pos ( gl n ). By definition, the basis B pos ( gl n ) is invariant under the shift maps ξ p for p ∈ Z .Summarizing we have the following. Proposition 5.16. There exists a positive basis B pos ( gl n ) for A ˙ U ( gl n ) (and also for ˙ U ( gl n ) ),which is induced from the canonical basis for A ˙ U ( sl n ) . It is clear that the transition matrix between the positive basis and the stably canonicalbasis of A ˙ U ( gl n ) is unitriangular.6. Canonical basis for ˙ U ı ( sl n ) for n even In this section, we shall construct the canonical basis of ˙ U ı ( sl n ) for n even with positivityproperties. This is achieved by relating to the case of ˙ U ( sl n ) for n odd studied in theprevious two sections with n = n − ≥ n even) . ı Schur algebra S ı ( n , d ) and the transfer map φ ıd + n ,d . Recall A S ( n, d ) from Sec-tion 4.1. We define A S ı ( n , d ) to be the A -submodule of A S ( n, d ) spanned by the standardbasis element [ A ] d , where A runs over the following subset of Ξ d in (4.1).Ξ ıd = { A ∈ Ξ d | a n +1 ,j = δ n +1 ,j , a i, n +1 = δ i, n +1 } . (6.1)Clearly, this is a subalgebra of A S ( n, d ) over A . Note that when the parameter v is special-ized at √ q , the algebra A S ı ( n , d ) coincides with the convolution algebra of pairs of n -steppartial isotropic flag in F d +1 q equipped with a fixed non-degenerate symmetric bilinear form.Moreover, the subset {{ A } d | A ∈ Ξ ıd } of the canonical basis of A S ( n, d ) is an A -basis of A S ı ( n , d ). Let S ı ( n , d ) = Q ( v ) ⊗ A A S ı ( n , d ) . Recall from (4.8), we have an algebra homomorphism S ( n + d, d ) → S ( n, d ) ⊗ S ( n, n ). Byrestricting to S ı ( n , d ), we obtain an algebra homomorphism∆ ı : S ı ( n , n + d ) → S ı ( n , d ) ⊗ S ( n , n ) , where we identify S ( n , n ) with the subalgebra in S ( n, n ) spanned by the elements [ A ] whoseentries in the ( n + 1)st rows and columns are zero. We refer to [FL15, Lemma 5.1.1] for amore explicit construction of ∆ ı , which is denoted e ∆ ı therein. Recall the sign homomorphism χ n from (4.7), and we define the transfer map φ ıd + n ,d : S ı ( n , d + n ) → S ı ( n , d ) to be thecomposition φ ıd + n ,d : S ı ( n , d + n ) ∆ ı −−−→ S ı ( n , d ) ⊗ S ( n , n ) ⊗ χ n −−−→ S ı ( n , d ) . We set I = I − E n +1 ,n +1 . By [FL15, Corollary 5.1.4], we have φ ıd + n ,d ( { X } d + n ) = ( { X − I } d , if X − I ∈ Ξ ıd , , otherwise . (6.2)for all matrices X ∈ Ξ ıd such that either one of the following matrices is diagonal: X , X − E θ n , n +2 , X − aE θi +1 ,i or X − aE θi,i +1 where a ∈ N , 1 ≤ i ≤ n − Remark 6.1. As we will show that if X is chosen such that X − aE θ n , n +2 is diagonal for a ≥ 2, the formula (6.2) fails to be true. This makes the construction of canonical basis for˙ U ı ( sl n ) more subtle than that of ˙ U ( sl n ). This subtlety boils down to the detailed analysisof the rank-one transfer map, which is the main topic of the following subsection.6.2. The transfer map on S (2 , d ) . In this subsection, we set n = 2 (hence r = 1) andconsider the rank-one transfer map φ ıd,d − : S ı (2 , d ) −→ S ı (2 , d − A a,b = a b b a . (6.3)Thus if A a,b ∈ Ξ d , we have a + b = d . In this subsection we drop the index d to write [ A a,b ]and { A a,b } for [ A a,b ] d and { A a,b } d , respectively. We set [ A a,b ] = 0 , if a < b < . Lemma 6.2. For all a, b ∈ N such that a + b = d , we have φ ıd,d − ([ A a,b ]) = [ A a − ,b ] + ( v − a +1 − v − a − )[ A a − ,b − ] − v − a − [ A a,b − ] . Proof. We shall prove the lemma by induction on b . When b = 0, the statement follows fromthe definition of φ ıd,d − .Let b ∈ N , and we assume the formula in the lemma is proved for φ ıd,d − ([ A a,b ′ ]), for all b ′ ≤ b and all a . We set t d = { A d − , } . Recall from [BKLW, Lemma A.13] that we have t d ∗ [ A a,b ] = v − a + b [ A a,b ] + v b [ b + 1][ A a − ,b +1 ] + v b − [ a + 1][ A a +1 ,b − ] . (6.4)By induction and using (6.4), we have φ ıd,d − ( t d ∗ [ A a,b ]) = φ ıd,d − ( v − a + b [ A a,b ] + v b [ b + 1][ A a − ,b +1 ] + v b − [ a + 1][ A a +1 ,b − ])= ( v − a + b +2 + v b − [ b ]( v − a +1 − v − a − ))[ A a − ,b ] + v b [ b + 1][ A a − ,b +1 ]+ (cid:16) v b − [ a − 1] + ( v − a +1 − v − a − ) v − a + b − v − a + b − [ b − (cid:17) [ A a − ,b − ]+ (cid:16) v b − [ a ]( v − a +1 − v − a − ) − v − a + b − (cid:17) [ A a,b − ] − v − a + b − [ a + 1][ A a +1 ,b − ] . (6.5) OSITIVITY VS NEGATIVITY OF CANONICAL BASES 27 By combining (6.4) and (6.5), we have v b [ b + 1] φ ıd,d − ([ A a − ,b +1 ])= φ ıd,d − ( t d ∗ [ A a,b ]) − v − a + b φ ıd,d − ([ A a,b ]) − v b − [ a + 1] φ ıd,d − ([ A a +1 ,b − ])= v b [ b + 1][ A a − ,b +1 ] + (cid:16) v − a + b +2 − v − a + b + v b − [ b ]( v − a +1 − v − a − ) (cid:17) [ A a − ,b ]+ (cid:16) v b − [ a − − v − a b − [ b − − v b − [ a + 1] (cid:17) [ A a − ,b − ]+ (cid:16) v b − [ a ]( v − a +1 − v − a − ) − v − a + b − + v − a + b − − v b − [ a + 1]( v − a − v − a − ) (cid:17) [ A a,b − ]= v b [ b + 1] (cid:0) [ A a − ,b +1 ] + ( v − a +2 − v − a )[ A a − ,b ] − v − a +1 [ A a − ,b − ] (cid:1) . Thus we have φ ıd,d − ([ A a − ,b +1 ]) = [ A a − ,b +1 ] + ( v − ( a − − v − ( a − − )[ A a − ,b ] − v − a − − [ A a − ,b − ] . The lemma is proved. (cid:3) Proposition 6.3. We have φ ıd,d − ( { A a,b } ) = { A a − ,b } , if a ≥ , { A ,b − } , if a = 1 , , if a = 0 . (6.6) Proof. The coefficients of [ A a − ,b − ] and [ A a,b − ] in the expansion of φ ıd,d − ([ A a,b ]) are in v − Z [ v − ] for a ≥ 2, by Lemma 6.2. Meanwhile, A a ′ ,b ′ (cid:22) A a,b if and only if a ′ ≥ a . So wehave φ ıd,d − ( { A a,b } ) ∈ [ A a − ,b ] + b X i =1 v − Z [ v − ][ A a − i,b − i ] . (6.7)Since φ ıd,d − ( { A a,b } ) is bar invariant, we conclude that φ ıd,d − ( { A a,b } ) = { A a − ,b } if a ≥ a = 1, we write { A ,b } = [ A ,b ] + b X i =1 Q i [ A i,b − i ] , for some Q i ∈ v − Z [ v − ] . Thus(6.8) φ ıd,d − ( { A ,b } ) = (1 − v − )[ A ,b − ] − v − [ A ,b − ] + b X i =1 Q i φ ıd,d − ([ A i,b − i ]) . By Lemma 6.2, the coefficient of [ A ,b − ] on the RHS of (6.8) is in 1 + v − Z [ v − ] and the coef-ficients of [ A i,b − i ] on the RHS of (6.8) for i ≥ v − Z [ v − ]. Now since φ ıd,d − ( { A a,b } )is bar invariant, the coefficient of [ A ,b − ] must be 1, and we have φ ıd,d − ( { A ,b } ) = { A ,b − } .Now Lemma 6.2 for a = 0 gives us φ ıd,d − ([ A ,b ]) = − v − [ A ,b − ]. A similar analysis as for a = 1 shows that the expansion of φ ıd,d − ( { A ,b } ) with respect to the standard basis [ A a,b ]have all coefficients in v − Z [ v − ]. This yields φ ıd,d − ( { A ,b } ) = 0 due to its bar-invarianceproperty.The proposition is proved. (cid:3) In Section 7.3, we will give an explicit formula of the canonical basis in S ı (2 , d ) in termsof standard basis.6.3. Hybrid monomial basis for S ı ( n , d ) . Now we consider S ı ( n , d ) for a general eveninteger n . Recall the monomial basis { d M A | A ∈ Ξ ıd } of S ı ( n , d ) from [BKLW, Proposition 5.6];for notation Ξ ıd see (6.1). This is a subset of the monomial basis { d M A } in S ( n, d ) [BKLW,(3.25)] (denoted by m A therein) used in Section 5.1, and d M A is a monomial in [ X ] d whereeither X − aE θi,i +1 , or X − aE θi +1 ,i , for all 1 ≤ i ≤ n − 1, is diagonal or a twin product[ X ] d ∗ [ Y ] d , where the matrices X − aE θ n , n +1 and Y − aE θ n +1 , n for some a ∈ N are diagonaland co( X ) = ro( Y ). Recall that the subset {{ A } d | A ∈ Ξ ıd } of the canonical basis of S ( n, d )forms a basis for S ı ( n , d ) and for the twin pair [ X ] d ∗ [ Y ] d , we have[ X ] d ∗ [ Y ] d = { E θ n , n +2 ( a ) } d + lower terms ∈ S ı ( n , d ) . (6.9)Here E θ n , n +2 ( a ) is the unique matrix defined by the conditions: co (cid:0) E θ n , n +2 ( a ) (cid:1) = co( Y ) and E θ n , n +2 ( a ) − aE θ n , n +2 is diagonal. Definition 6.4. The hybrid monomial d M ıA is obtained from d M A by replacing the twinproduct [ X ] d ∗ [ Y ] d in d M A by the leading term { E θ n , n +2 ( a ) } d in (6.9).The following properties of the hybrid monomials d M ıA are the main reasons to introducethem. Proposition 6.5. The following properties hold for a hybrid monomial d M ıA (where A ∈ Ξ ıd ): (1) d M ıA = d M ıA , (2) d M ıA = { A } d + lower term, (3) the set { d M ıA (cid:12)(cid:12) A ∈ Ξ ıd } forms a basis of A S ı ( n , d ) , (4) φ ıd,d − n ( d M ıA ) = d − n M ıA , whenever a ii ≫ for all i ∈ [1 , n ] .Proof. Items (1)-(3) follow readily by construction. Since d M ıA is obtained by modifying thefactors in d M A at finitely many places, it is clear that we can add p I for p large enough to A such that all twin product [ X ] d ∗ [ Y ] d appearing in d M A + p I have their ( n , n )th entries ≥ (cid:3) The modified quantum coideal subalgebras ˙ U ı ( gl n ) and ˙ U ı ( sl n ) . Recall the alge-bra K from Section 4.1. This algebra has a standard basis [ A ] parameterized by the set ˜Ξin (4.2). Let K be the subalgebra of K spanned by the standard basis [ A ] in ˜Ξ such thatro( A ) n +1 = co( A ) n +1 = 1. Let J be the ideal of K spanned by [ A ] for all A ∈ ˜Ξ ı such that a n +1 , n +1 < 0. Let ˜Ξ ı be the subset of ˜Ξ consisting of matrices A defined by a n +1 ,j = δ n +1 ,j and a i, n +1 = δ i, n +1 for all i, j .We set K ı be the quotient of K by J . It is shown in [BKLW, Appendix A.3] that K ı admits a monomial basis M A + J , a standard basis [ A ] + J , and a canonical basis { A } + J ,for all A ∈ ˜Ξ ı . Furthermore, it is shown in [BKLW, Proposition A.11] that K ı is isomorphicto the modified quantum coideal subalgebra ˙ U ı ( gl n ) of the quantum algebra U ( gl n ). Weshall identify K ı with ˙ U ı ( gl n ). Recall that the algebra ˙ U ı ( gl n ) is an associative Q ( v )-algebragenerated by the symbols 1 λ , e i λ , 1 λ e i , f i λ , 1 λ f i , t λ , and 1 λ t , for i = 1 , . . . , n − OSITIVITY VS NEGATIVITY OF CANONICAL BASES 29 λ ∈ Z ı n := { λ ∈ Z n | λ n +1 = 1 } , subject to the following relations (6.10): for i, j = 1 , . . . , n − λ, λ ′ ∈ Z ı n , and for x, x ′ ∈ { , e i , e j , f i , f j , t } , x λ λ ′ x ′ = δ λ,λ ′ x λ x ′ ,e i λ = 1 λ − α i e i ,f i λ = 1 λ + α i f i ,t λ = 1 λ t,e i λ f j = f j λ − α i − α j e i , if i = j,e i λ f i = f i λ − α i e i + [ λ i +1 − λ i ]1 λ − α i , ( e i e j + e j e i )1 λ = [2] e i e j e i λ , if | i − j | = 1 , ( f i f j + f j f i )1 λ = [2] f i f j f i λ , if | i − j | = 1 ,e i e j λ = e j e i λ , if | i − j | > ,f i f j λ = f j f i λ , if | i − j | > ,tf i λ = f i t λ , if i = n − , ( t f n − + f n − t )1 λ = (cid:0) [2] tf n − t + f n − (cid:1) λ , ( f n − t + tf n − )1 λ = [2] f n − tf n − λ ,te i λ = e i t λ , if i = n − , ( t e n − + e n − t )1 λ = (cid:0) [2] te n − t + e n − (cid:1) λ , ( e n − t + te n − )1 λ = [2] e n − te n − λ . (6.10)Here λ ± α i are the short hand notations introduced in Section 4.1. To simplify the notation,we shall write x λ · x λ · · · x l λ l = x x · · · x l λ l , if the product is not zero.We define an equivalence relation ≈ on Z ı n by setting λ ≈ λ ′ if and only if λ − λ ′ = a I forsome a ∈ Z . Let ˆ Z ı n be the set Z ı n / ≈ of equivalence classes. Let ˙ U ı ( sl n ) be the algebradefined in the same fashion as ˙ U ı ( gl n ) with the parameter set Z ı n replaced by ˆ Z ı n . Similar to˙ U ( gl n ), the algebras ˙ U ı ( gl n ) and ˙ U ı ( sl n ) admit the following decompositions.˙ U ı ( gl n ) = M d ∈ Z ˙ U ı ( gl n ) h d i , ˙ U ı ( sl n ) = M ¯ d ∈ Z / n Z ˙ U ı ( sl n ) h ¯ d i , where ˙ U ı ( gl n ) h d i is spanned by elements of the form 1 λ u µ with | µ | = | λ | = 2 d + 1 and u ∈ ˙ U ı ( gl n ), and ˙ U ı ( sl n ) h ¯ d i is spanned by 1 µ ˙ U ı ( sl n )1 λ , where µ, λ ∈ ˆ Z ı n , | µ | ≡ | λ | ≡ d + 1mod 2 n .We have the following commutative diagram similar to (4.13):˙ U ı ( gl n ) ℘ ı (cid:15) (cid:15) Ψ ıd + n | | ②②②②②②②②②②②②②②②②②②②②②②② Ψ ıd ! ! ❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇ ˙ U ı ( sl n ) φ ıd + n u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ φ ıd ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ S ı ( n , d + n ) φ ıd + n ,d / / S ı ( n , d )(6.11) Here the homomorphisms φ ıd and ℘ ı are defined in a similar way as φ d and ℘ in (4.13)respectively, but with I replaced by I .6.5. Inner product on ˙ U ı ( sl n ) . Let h− , −i ı,d be the bilinear form on the ı Schur alge-bra S ı ( n , d ) obtained from the bilinear form h− , −i d on S ( n, d ) by restriction, thanks to S ı ( n , d ) ⊂ S ( n, d ). We define a family of bilinear forms h− , −i ı,d on ˙ U ı ( sl n ) by pullingback the one on the Schur algebra level via φ ıd in (6.11), i.e., h x, x ′ i ı,d = h φ ıd ( x ) , φ ıd ( x ′ ) i ı,d for x, x ′ ∈ ˙ U ı ( sl n ). We shall study the behavior of these bilinear forms as d tends to infinity.We need the following analogue of [M12, Lemma 4.2]. Lemma 6.6. Let A i ( ≤ i ≤ k ) be matrices such that either A i − E θh +1 ,h , A i − E θh,h +1 ,( h ∈ [1 , n − ), or A i − E θ n , n +2 is diagonal. Let A ∈ ˜Ξ ı with | A | = d . Then there existsmatrices Z , Z , · · · , Z m ∈ ˜Ξ ı , and G ( v, u ) , · · · , G m ( v, u ) ∈ Q ( v )[ u ] and an integer p ∈ Z such that { A + p I } d + p n ∗ { A + p I } d + p n ∗ · · · ∗ { A k + p I } d + p n ∗ [ A + p I ] d + p n = m X i =1 G i ( v, v − p )[ Z i + p I ] d + p n , for all even integer p ≥ p .Proof. The proof follows the arguments of [M12, Lemma 4.2] and [BKLW, Lemma A.1],except that we need to take care of the new case when k = 1 and A − E θ n , n +2 is diagonal. Inthis case, we need the following multiplication formula in S ı ( n , d ) from [BKLW, Lemma A.13]for the generator t d = P { X } d where X runs over all matrices in Ξ ıd such that X − E θ n , n +2 is diagonal. (That t d can be written in such a form is due to [BKLW, Lemma 5.5].) For any A ∈ Ξ ıd , we have t d ∗ [ A ] d = X ≤ j ≤ n v P j ≥ p a n ,p − P j>p a n ,p − P p> n δ j,p [ a n +2 ,j + 1][ A − E θ n ,j + E θ n +2 ,j ] d . (6.12)Then we set Z j = A − E θ n ,j + E θ n +2 ,j and G j ( v, u ) = v P j ≥ p a n ,p − P j>p a n ,p − P p> n δ j,p u δ j, n v − a n ,j +1) u δ j, n − v − − . The lemma now follows by induction. (cid:3) Remark 6.7. Note that in the multiplication formula in Lemma 6.6, the canonical basiselements are used instead of the standard basis elements, which are the same for all generatorsexcept t d .We are ready to state the asymptotic behavior of the form h· , ·i ı,d . Proposition 6.8. As p goes to infinity, the limit lim p →∞ h x, x ′ i ı,d + p n , for all x, x ′ ∈ ˙ U ı ( sl n ) ,converges in Q (( v − )) to an element in Q ( v ) .Proof. The proof is similar to [M12]. We need the adjointness of the bilinear form h· , ·i ı,d ,which is inherited from that of h· , ·i d , and in particular we have h t d ∗ { A } d , { B } d i ı,d = h{ A } d , t d ∗ { B } d i ı,d OSITIVITY VS NEGATIVITY OF CANONICAL BASES 31 from [BKLW, Corollary 3.15]. The only difference from [M12] is that we work in a largerring Q ( v )[ u ]. Now suppose that G ( v, u ) = P ni =0 a i u i where a i ∈ Q ( v ). Then we have G ( v, v − p ) = m X i =0 a i v − pi , which implies that lim p →∞ G ( v, v − p ) = a in Q (( v − )). (cid:3) Similar to the form h· , ·i , we define a bilinear form h− , −i ı on ˙ U ı ( sl n ) (independent of d )by letting h x, x ′ i ı = n − X d =0 lim p →∞ h x, x ′ i ı,d + p n , ∀ x, x ′ ∈ ˙ U ı ( sl n ) . (6.13)6.6. Hybrid monomial basis in ˙ U ı ( gl n ) . As for the construction of the canonical basis for˙ U ( sl n ), we need a version of monomial basis on ˙ U ı ( gl n ) which enjoys similar properties inProposition 6.5 in order to construct the canonical basis of ˙ U ı ( sl n ). We lift the basis { d M ıA } of S ı ( n , d ) to a basis of ˙ U ı ( gl n ) with the desired properties. The procedure is exactly thesame as used in the construction of the basis { d M ıA } for the ı Schur algebras in Section 6.3.More precisely, recall a monomial basis { M A | A ∈ ˜Ξ ı } for K ı ≡ ˙ U ı ( gl n ) was constructed in[BKLW, Appendix A] by lifting the (usual) monomial basis { d M A } for ı Schur algebras. weform the hybrid monomial M ıA from M A by substituting any twin product [ X ] ∗ [ Y ] in M A asin (6.9) with its leading term { E θ n , n +2 ( a ) } with indices d dropped. Proposition 6.9. The following properties hold for a hybrid monomial M ıA with A ∈ ˜Ξ ı : (1) M ıA = M ıA , (2) M ıA = { A } + lower term, (3) the set { M ıA | A ∈ ˜Ξ ı } forms a basis of A ˙ U ı ( gl n ) , (4) φ ıd ( M ıA ) = d M ıA , whenever a ii ≫ for all ≤ i ≤ n .Proof. All properties follow readily from the constructions except the last one. As thehybrid monomial bases for ˙ U ı ( sl n ) and S ı ( n , d ) are defined multiplicatively by the sameprocedure, we only need to show that Property (4) in the rank one case. We remind that byconstruction the hybrid monomial basis in the rank one case is exactly the canonical basis.Hence Property (4) at rank one is exactly the statement of [BKLW, Proposition A.21]. (cid:3) Now since we have Proposition 6.5, the commutative diagram (6.11), Proposition 6.9 athand, the constructions and results in Sections 5 in the -setting can be rerun for the ı -setting.Let us outline them. Proposition 6.10. Given A ∈ ˜Ξ ı , we have ξ ı − ( { p I A } ) = { ( p − I A } , ℘ ı ( { p I A } ) = ℘ ı ( { ( p − I A } ) , for all even integer p ≫ , where p I A = A + p I .Proof. The same type arguments of the proof of Proposition 5.3 work here. (cid:3) We define an equivalence on ˜Ξ ı by A ≈ B if and only if A − B = p I for some even integer p . We set ˆΞ ı = ˜Ξ ı / ≈ . By Proposition 6.10, the following definition is well defined. Definition 6.11. We define b ˇ A = ℘ ı ( { p I A } ) ∈ ˙ U ı ( sl n ), ∀ p ≫ , ˇ A ∈ ˆΞ ı .Now as we have the key properties established in Propositions 6.9–6.10, we are in a posi-tion to establish the ı -counterparts of results on canonical bases in Sections 5.2–5.3, whosesimilar proofs will be skipped. Below is a summary of the ı -counterparts of Theorem 5.5,Theorem 5.6, Proposition 5.7, and Theorem 5.8. Theorem 6.12. (1) The set B ı ( sl n ) = { b ˇ A | ˇ A ∈ ˆΞ ı } forms a basis for ˙ U ı ( sl n ) and for A ˙ U ı ( sl n ) . (2) The structure constants for the algebra ˙ U ı ( sl n ) with respect to the basis B ı ( sl n ) arepositive (i.e., in N [ v, v − ] ). (3) The form h− , −i ı on ˙ U ı ( sl n ) is non-degenerate. Moreover, the basis B ı ( sl n ) is almostorthonormal and positive with respect to this form, i.e., h b ˇ A , b ˇ B i ı ∈ δ ˇ A, ˇ B + v − N [[ v − ]] . Again, similar to Proposition 5.9, the signed canonical basis − B ı ( sl n ) ∪ B ı ( sl n ) is charac-terized by the bar invariance, integrality and almost orthonormality. Remark 6.13. The main results (Theorem 5.10, Proposition 5.11, Proposition 5.13, Corol-lary 5.14, Propositions 5.15–5.16) in Sections 5.4–5.6 admit ı -analogues here with n replacedby n and by ı , respectively. Remark 6.14. Shigechi [Sh14] has established by combinatorial methods certain positivityof the ı -canonical bases (introduced in [BW13]) on general tensor products of modules of thequantum coideal algebra of U ( sl ), and this supports our general positivity conjectures. SeeRemark 6.13 for a closely related result.7. Formulas of canonical basis of S (2 , d )7.1. Combinatorial identities. Recall the quantum v -binomial coefficients were definedin (2.3) for m ∈ Z and b ∈ N . We introduce the following additional notation (cid:20) mb (cid:21) v = Y ≤ i ≤ b v m − i +1) − v i − . We first establish two combinatorial identities which are needed in later computations andcould be of some independent interest as well. Lemma 7.1. For any a ∈ Z and p ∈ N , we have p X s =0 v s ( a +2 s ) (cid:20) ps (cid:21) v p − s Y k =1 (1 − v a +4 s +4 k ) = 1 . Proof. Recall the quantum binomial identity(7.1) (cid:20) ps (cid:21) v = (cid:20) p − s (cid:21) v + v p − s (cid:20) p − s − (cid:21) v . We prove the lemma by induction on p , with the base case for p = 0 being trivial.By (7.1), we can rewrite the sum as a sum of two summands: p X s =0 v s ( a +2 s ) (cid:20) ps (cid:21) v p − s Y k =1 (1 − v a +4 s +4 k ) = S + S , OSITIVITY VS NEGATIVITY OF CANONICAL BASES 33 where S = p X s =1 v s ( a +2 s ) v p − s (cid:20) p − s − (cid:21) v p − s Y k =1 (1 − v a +4 s +4 k ) ,S = p − X s =0 v s ( a +2 s ) (cid:20) p − s (cid:21) v p − s Y k =1 (1 − v a +4 s +4 k ) . Setting p ′ = p − , s ′ = s − a ′ = a + 2, we have a + 2 s = a ′ + 2 s ′ , and thus by theinductive assumption (with p ′ < p ) we obtain S = v a +4 p p ′ X s ′ =0 v s ′ ( a ′ +2 s ′ ) (cid:20) p ′ s ′ (cid:21) v p ′ − s ′ Y k =1 (1 − v a ′ +4 s ′ +4 k ) = v a +4 p . Setting p ′ = p − 1, by the inductive assumption (with p ′ < p ) again we have S = p ′ X s =0 v s ( a +2 s ) (cid:20) p ′ s (cid:21) v p ′ − s Y k =1 (1 − v a +4 s +4 k ) · (1 − v a +4 p ) = 1 − v a +4 p . Summing up S and S above we have proved the lemma. (cid:3) Lemma 7.2. For m ∈ N , we have m X j =0 v ( m − j )( m − j +1) Q ju =1 (1 − v m − u +1) ) Q ⌊ j ⌋ k =1 (1 − v k ) = 1 . Proof. Set m = 2 n if m is even or m = 2 n + 1 otherwise. We first sum up the two summandswith j = 2 d and j = 2 d + 1, for fixed d with 0 ≤ d ≤ n : v ( m − d )( m − d +1) Q du =1 (1 − v m − u +1) ) Q dk =1 (1 − v k ) + v ( m − d − m − d ) Q d +1 u =1 (1 − v m − u +1) ) Q dk =1 (1 − v k )= v ( m − d − m − d ) Q du =1 (1 − v m − u +1) ) Q dk =1 (1 − v k )= v ( m − d − m − d ) (cid:20) nd (cid:21) v d Y k =1 (1 − v n − d +4 k ∓ ) , where the sign ‘ − ’ is always taken for m = 2 n and ‘+’ for m = 2 n + 1 on the right-handside above and similar places below. Note that the above is actually valid for d = n in case m = 2 n as well, where the second summand on the left-hand side is simply zero. Hence, noting (cid:20) nd (cid:21) v = (cid:20) nn − d (cid:21) v and setting s = n − d , we have m X j =0 v ( m − j )( m − j +1) Q ju =1 (1 − v m − u +1) ) Q ⌊ j ⌋ k =1 (1 − v k )= n X d =0 v (2 n − d ∓ n − d ) (cid:20) nd (cid:21) v d Y k =1 (1 − v n − d +4 k ∓ )= n X s =0 v s (2 s ∓ (cid:20) ns (cid:21) v n − s Y k =1 (1 − v s +4 k ∓ ) = 1 , where the last equation uses Lemma 7.1 (where we set a = ∓ p = n ). The lemma isproved. (cid:3) The bar conjugate of the standard basis. Let T = G d ≥ S ı (2 , d )be the Q ( v )-vector space with the standard basis { [ A a,r ] | a, r ∈ N } . As before we set [ A a,r ] =0 if a < r < . We introduce a shorthand notation to denote the monomial basis element M a,r = d M A a,r . By [BKLW, (5.4)], we have M a,r = [ A a,r ] + r X i =1 v β a ( i ) (cid:20) a + ii (cid:21) [ A a + i,r − i ](7.2)where(7.3) β a ( i ) = ai − i ( i + 1) . Then { M a,r | a, r ∈ N , a + b = d } forms a monomial basis for S ı (2 , d ), and so { M a,r | a, r ∈ N } forms a monomial basis for T . There is a Q -linear bar involution on S ı (2 , d ) for all d andhence on T , denoted by , which fixes each M a,r . Note that(7.4) (cid:20) ma (cid:21) = v a ( a − m ) (cid:20) ma (cid:21) , and M a,r = M a,r . The following theorem is obtained with help from a UVA undergraduate Tahseen Rab-bani (supported by NSF), whose computer computation for small values of r was crucial informulating the precise statement. Theorem 7.3 (joint with Tahseen Rabbani) . For all a, r ∈ N , we have [ A a,r ] = r X i =0 v − ia − ( i +12 ) · Q ik =1 (1 − v a + k ) ) Q ⌊ i ⌋ k =1 (1 − v k ) [ A a + i,r − i ]= r X i =0 Q ik =1 ( v − a − k − v a + k ) Q ⌊ i ⌋ k =1 (1 − v k ) [ A a + i,r − i ] . OSITIVITY VS NEGATIVITY OF CANONICAL BASES 35 Proof. The two expressions in the statement are clearly equal. We shall proceed by inductionon r . The base case for r = 0 is clear.Assume the formula is verified for [ A a,r ′ ] for all a, r ′ ∈ N such that r ′ < r . By (7.2) and M a,r = M a,r , it suffices to verify the formula for [ A a,r ] as given in the theorem satisfies that[ A a,r ] + r X i =1 v − β a ( i ) (cid:20) a + ii (cid:21) [ A a + i,r − i ] = [ A a,r ] + r X i =1 v β a ( i ) (cid:20) a + ii (cid:21) [ A a + i,r − i ]Equating the coefficients of [ A a + m,r − m ] on both sides of the above identity, we are reducedto verifying the following identity for 0 ≤ m ≤ r : X i + j = m v − ai + i ( i +1) (cid:20) a + ii (cid:21) Q jk =1 ( v − a − i − k − v a + i + k ) Q ⌊ j ⌋ k =1 (1 − v k ) = v − am − m ( m +1) (cid:20) a + mm (cid:21) . (We have used (7.4) on deriving the right-hand side above.)After further simplification using (cid:20) a + ii (cid:21) = [ a + i ]![ a ]![ i ]! and i = m − j , the above identity isreduced to the following identity for m ≥ m X j =0 v ( m − j )( m − j +1) [ a + m − j ]![ m − j ]! Q ju =1 [ a + m + 1 − u ] · (1 − v ) j Q ⌊ j ⌋ k =1 (1 − v k ) = [ a + m ]![ m ]! . Thanks to [ a + m − j ]! Q ju =1 [ a + m + 1 − u ] = [ a + m ]!, the above identity is equivalent tothe identity in Lemma 7.2. The theorem is proved. (cid:3) Denote the coefficient of A a + i,r − i in Theorem 7.3 above, which is independent of r , by(7.5) b ia = v − ia − ( i +12 ) · Q ik =1 (1 − v a + k ) ) Q ⌊ i ⌋ k =1 (1 − v k ) , with b a = 1 . Then we have(7.6) [ A a,r ] = r X i =0 b ia · [ A a + i,r − i ] , for all a, r ∈ N . Example 7.4. For a ∈ N , we have[ A a, ] = [ A a, ] , [ A a, ] = [ A a, ] + ( v − a − − v a +1 )[ A a +1 , ] , [ A a, ] = [ A a, ] + ( v − a − − v a +1 )[ A a +1 , ] + v − a − (1 − v a +1) )(1 − v a +2) )1 − v [ A a +2 , ] . Formulas for canonical basis of S ı (2 , d ) . The canonical basis is the Q ( v )-basis {{ A a,r }| a, r ∈ N } for T , which is completely determined by the bar invariance togetherwith the following property: { A a,r } = [ A a,r ] + r X i =1 γ a,r ( i )[ A a + i,r − i ] , for γ a,r ( i ) ∈ v − Z [ v − ] . (7.7)We denote γ a,r (0) = 1. Lemma 7.5. The polynomials γ a,r ( i ) are independent of r ; we shall write γ a ( i ) = γ a,r ( i ) . Proof. We shall show by induction on i ≥ 0. The case for i = 0 is clear.By (7.7), we have P ri =0 γ a,r ( i )[ A a + i,r − i ] = P rj =0 γ a,r ( j )[ A a + j,r − j ] . Equating the coefficientsof [ A a + i,r − i ] on both sides of this equation with the help of (7.6) gives us(7.8) γ a,r ( i ) − γ a,r ( i ) = i − X j =0 γ a,r ( j ) b i − ja + j . It follows from this and an easy induction on i that γ a,r ( i ) is independent of r . (cid:3) The next theorem establishes formulas for the canonical basis { A a,r } for S ı (2 , d ) for all d ,or equivalently by (7.7), determines γ a ( i ) for all a, i ∈ N . Theorem 7.6. (1) For a, s ∈ N with a even, we have γ a (2 s ) = v − s − s s Y k =1 − v − a − k − v − k ,γ a (2 s + 1) = v − a − s − s − s Y k =1 − v − a − k − v − k . (2) For a, s ∈ N with a odd, we have γ a (2 s ) = v − s + s s Y k =1 − v − a − k − − v − k ,γ a (2 s + 1) = v − a − s − s − s Y k =1 − v − a − k − − v − k . In other words, these polynomials γ a ( r ) are all essentially v -binomial coefficients. Proof. Let us rewrite (7.8) as(7.9) γ a ( r ) = r X i =0 γ a ( i ) b r − ia + i . This formula uniquely determines the polynomials γ a ( r ) for all a, r ∈ N (by induction on r ), which satisfy γ a (0) = 1 and γ a ( r ) ∈ v − Z [ v − ] for r ≥ 1. It suffices to verify that theformulas for γ a ( r ) given in the theorem do satisfy (7.9). The verification is divided into 4very similar cases, depending on the parity of a and the parity of r .Assume first that both a and r are odd. Set r = 2 p + 1. Let 0 ≤ s ≤ p . We have γ a (2 s ) b r − sa +2 s = v s ( a +2 s ) − ar − ( r +12 ) s Y k =1 − v a +4 k +2 − v k · Q r − su =1 (1 − v a +4 s +2 u ) Q ⌊ r ⌋− sk =1 (1 − v k ) ,γ a (2 s + 1) b r − s − a +2 s +1 = v s ( a +2 s ) − ar − ( r +12 ) +2 a +4 s +2 s Y k =1 − v a +4 k +2 − v k · Q r − s − u =1 (1 − v a +4 s +2 u +2 ) Q ⌊ r − ⌋− sk =1 (1 − v k ) . OSITIVITY VS NEGATIVITY OF CANONICAL BASES 37 The above two formulas have almost identical factors except that γ a (2 s ) b r − sa +2 s has an extrafactor (1 − v a +4 s +2 ) while γ a (2 s + 1) b r − s − a +2 s +1 has an extra factor v a +4 s +2 . Hence, r X i =0 γ a ( i ) b r − ia + i = p X s =0 (cid:16) γ a (2 s ) b r − sa +2 s + γ a (2 s + 1) b r − s − a +2 s +1 (cid:17) = p X s =0 v s ( a +2 s ) − ar − ( r +12 ) Q sk =1 (1 − v a +4 k +2 ) Q sk =1 (1 − v k ) · Q p − su =1 (1 − v a +4 s +2 u +2 ) Q p − sk =1 (1 − v k ) . Using p − s Y u =1 (1 − v a +4 s +2 u +2 ) = p Y k = s +1 (1 − v a +4 k +2 ) · p − s Y k =1 (1 − v a +4 s +4 k ) , we can rewrite the above equation as r X i =0 γ a ( i ) b r − ia + i = v − ar − ( r +12 ) p X s =0 v s ( a +2 s ) Q pk =1 (1 − v a +4 k +2 ) Q sk =1 (1 − v k ) Q p − sk =1 (1 − v k ) · p − s Y k =1 (1 − v a +4 s +4 k ) . (7.10)On the other hand, we have(7.11) γ a ( r ) = v − ar − ( r +12 ) Q pk =1 (1 − v a +4 k +2 ) Q pk =1 (1 − v k ) . 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