Serre-Lusztig relations for \imathquantum groups II
aa r X i v : . [ m a t h . QA ] F e b SERRE-LUSZTIG RELATIONS FOR ı QUANTUM GROUPS II
XINHONG CHEN, MING LU, AND WEIQIANG WANG
Abstract.
The ı Serre relations and the corresponding Serre-Lusztig relations are formu-lated for arbitrary ı quantum groups arising from quantum symmetric pairs of Kac-Moodytype. This generalizes the main results in [CLW18, CLW20]. Introduction ı Serre relations in [CLW18]and the corresponding Serre-Lusztig (or higher order Serre) relations in [CLW20] amongChevalley generators B i and B j , for τ i = i = w • i and i = j ∈ I ◦ , in ı quantum groups U ı or e U ı arising from quantum symmetric pairs of arbitrary Kac-Moody type, to the general casesfor τ i = i (with condition w • i = i dropped). The notations are to be explained below, andwe refer to loc. cit. for a more complete introduction of backgrounds.1.2. We are concerned about the Serre type relations among the generators B i in an ı quantum group U ı (or a universal ı quantum group e U ı ) arising from quantum symmetricpairs (QSP) ( U , U ı ). Recall that the definition of QSP is built on the Satake diagrams or ad-missible pairs ( I = I ◦ ∪ I • , τ ) [Le02, Ko14], and the universal QSP ( e U , e U ı ) appeared in [LW19].The Serre type relations of U ı are obtained by G. Letzter [Le02] in finite type. The Serrerelations between B i , B j (where B j appears in degree 1) were explicitly known [Ko14, BK15]in an arbitrary ı quantum group U ı (or e U ı ) of Kac-Moody type, unless τ i = i ∈ I ◦ ; in casewhen τ i = i ∈ I ◦ , explicit Serre relations were written down under a strong constraint onthe Cartan integers | a ij | ≤
3, cf. [BK15]. General ı Serre relations for τ i = i = w • i ∈ I ◦ and j ∈ I ◦ without any constraint on Cartan integers a ij have been formulated by the authors[CLW18]; see (1.2) below. “Explicit” yet unwieldy formulas for Serre relations in an arbi-trary ı quantum group U ı are also obtained in [DeC19]; the coefficients involved therein canbe rather difficult to compute in practice.The Serre-Lusztig (or higher order Serre) relations for e U ı hold in closed forms [CLW20] e y i,j ; n,m,p,t,e = 0 , (1.1)for τ i = i = w • i ∈ I ◦ , i = j ∈ I ◦ and m ≥ − na ij ; see (3.3)–(3.4) for the definition of e y .These are generalizations of higher order Serre relations for quantum groups in [Lus94]. Incase n = 1 and m = 1 − a ij , the above relation reduces to the ı Serre relation [CLW18] − a ij X r =0 ( − r B ( r ) i,a ij + p i B j B (1 − a ij − r ) i,p i = 0 , (1.2)which hold for τ i = i = w • i ∈ I ◦ , i = j ∈ I ◦ in an arbitrary ı quantum group; more generally,the Serre-Lusztig relations of minimal degree (i.e., (1.1) for m = 1 − na ij and n ≥
1) takea similar simple form as in (1.2). These relations are expressed in terms of ı divided powers B ( m ) i,p introduced in [BW18a] and generalized in [BeW18, CLW20], depending on a parity p ∈ { ¯0 , ¯1 } .In particular, these relations hold for i = τ i in an arbitrary quasi-split ı quantum groups(i.e., when I • = ∅ ). Conjectures and examples for Serre-Lusztig relations of minimal degreesin ı quantum groups of split affine ADE type in very different forms were proposed earlierby Baseilhac and Vu [BaV14, BaV15]; their conjecture was proved for q -Onsager algebra in[Ter18].1.3. In our formulation, the ı Serre relations for an ı quantum group of arbitrary Kac-Moodytype formally takes the same form as (1.2): − a ij X r =0 ( − r B ( r ) i,a ij + p i B j B (1 − a ij − r ) i,p i = 0 , (1.3)where we have to replace B ( m ) i,p in (1.2) by a more general definition of ı -divided powers B ( m ) i,p defined in (2.3)–(2.4) (that is, ς i in B ( m ) i,p is replaced by ς i r i ( T w • E i )); in case when w • i = i , B ( m ) i,p is reduced to the original B ( m ) i,p thanks to r i ( E i ) = 1. Moreover, the Serre-Lusztigrelations (1.1) are generalized accordingly to arbitrary ı quantum group e U ı (see Theorem 3.8)and they follow by a recursive relation similar to the one in [CLW20] (see Theorem 3.7).For i ∈ I ◦ with τ i = i = w • i , a version of ı divided powers B ( m ) i (independent of p ∈ Z )were introduced in [BW18b] as a key ingredient toward ı canonical basis. These B ( m ) i (forsome suitable parameter ς i ) therein satisfies a crucial integral property, i.e., it lies in the Z [ q, q − ]-form of the modified ı quantum group. In contrast, the B ( m ) i,p introduced in thispaper are not integral for m ≥ ς i . Paraphrasing, the ı dividedpowers for τ i = i ∈ I ◦ , arising in 2 totally different settings of ı canonical basis and ı Serrerelations, miraculously coincide if and only if w • i = i . (Alas, we had a mental block on the“only if” part, and this explains why the formulation of this note were not noticed earlierwhen we were writing [CLW20].)1.4. The proof of the ı Serre and Serre-Lusztig relations in this note relies on the results in[CLW20] in an essential way, and a reader is recommended to keep a copy of it at hand. Inaddition, the proof here relies on another key ingredient, which we refer to as a universalityresult for ı quantum groups; see Proposition 2.2. This universality result was almost explicitin [Le02, Ko14] based on their projection techniques, and has been made very explicit in[DeC19]. While these authors were only concerned about Serre relations, the universalityresult applies well to Serre-Lusztig relations.The results of this paper bring us another step closer to have a Serre presentation of anarbitrary ı quantum group with all relations in neat closed forms; see Remark 3.3. Acknowledgement.
XC is supported by the Fundamental Research Funds for the CentralUniversities grant No. 2682020ZT100. WW is partially supported by the NSF grant DMS-2001351.
ERRE-LUSZTIG RELATIONS FOR ı QUANTUM GROUPS II 3 ı Divided powers and universality in ı quantum groups New ı divided powers for e U ı . Let ( a ij ) i,j ∈ I be a generalized Cartan matrix, and the as-sociated Drinfeld-Jimbo quantum group U = U I is a Q ( q )-algebra generated by E i , F i , K ± i ,for i ∈ I . The Drinfeld double e U = e U I is a Q ( q )-algebra generated by E i , F i , e K i , e K ′ i , for i ∈ I , and e K i e K ′ i is central in e U . Then U is obtained from e U by a central reduction: U = e U / ( e K i e K ′ i − | i ∈ I ). Let e U + (and respectively, U + ) be the subalgebra of e U (andrespectively, U ) generated by E i ( i ∈ I ). Clearly, e U + ∼ = U + , and we shall identify them. For i ∈ I , denote by r i : U + → U + the unique Q ( q )-linear maps [Lus94] such that r i (1) = 0 , r i ( E j ) = δ ij , r i ( xx ′ ) = xr i ( x ′ ) + q i · µ ′ r i ( x ) x ′ , (2.1)for x ∈ U + µ and x ′ ∈ U + µ ′ .Let ( I = I ◦ ∪ I • , τ ) be an admissible pair; cf. [Ko14, Definition 2.3]. Note that e U I • (and respectively, U I • ) is naturally a subalgebra of e U (and respectively, U ). The ı quantumgroup e U ı is a (coideal) subalgebra of e U (see [LW19, CLW20]), which is generated by e U I • , e k i = e K i e K ′ τi ( i ∈ I ◦ ), and B i = F i + T w • ( E τi ) e K ′ i ( i ∈ I ◦ ) . In this note, we are mainly concerned about B i for i ∈ I ◦ with τ i = i ; in this case, B i = F i + T w • ( E i ) e K ′ i ∈ e U ı . It is known (cf. [Ko14]) that[ r i ( T w • E i ) , B j ] = 0 , for i, j ∈ I ◦ . (2.2)For m ∈ Z , let [ m ] i = [ m ] q i denotes the quantum integer associate to q i . Let i ∈ I ◦ with τ i = i (but we drop the assumption that w • i = i which was imposed in [CLW18, CLW20]).The ı divided powers of B i in e U ı are defined to be B ( m ) i, ¯1 = 1[ m ] i ! B i Q kj =1 (cid:16) B i − [2 j − i q i e k i r i ( T w • E i ) (cid:17) if m = 2 k + 1 , Q kj =1 (cid:16) B i − [2 j − i q i e k i r i ( T w • E i ) (cid:17) if m = 2 k ;(2.3) B ( m ) i, ¯0 = 1[ m ] i ! B i Q kj =1 (cid:16) B i − [2 j ] i q i e k i r i ( T w • E i ) (cid:17) if m = 2 k + 1 , Q kj =1 (cid:16) B i − [2 j − i q i e k i r i ( T w • E i ) (cid:17) if m = 2 k. (2.4)Given p ∈ Z = { ¯0 , ¯1 } , the ı divided powers are determined by the following recursiverelations, for m ≥ B i B ( m ) i,p = ( [ m + 1] i B ( m +1) i,p if p = m, [ m + 1] i B ( m +1) i,p + [ m ] i q i e k i r i ( T w • E i ) B ( m − i,p if p = m. (2.5)We set B ( m ) i, ¯0 = 0 = B ( m ) i, ¯1 for any m < Remark . In case when w • i = i , we have r i ( T w • E i ) = 1, and the ı divided powers B ( m ) i,p were introduced first in [BW18a, BeW18] for a distinguished parameter ς i = q − i and theyare ı canonical basis elements in the modified ı quantum group. The B ( m ) i,p when w • i = i for XINHONG CHEN, MING LU, AND WEIQIANG WANG a general parameter ς i used in [CLW18, CLW20] (denoted by B ( m ) i,p ) are obtained from thedistinguished case above by a renormalization automorphism of U ı .In case when w • i = i and then r i ( T w • E i ) = 1, B ( m ) i,p in general do not lie in the Z [ q, q − ]-formof U ı . Toward the construction of ı canonical basis, different ı divided powers, B ( m ) i , whichlie in the Z [ q, q − ]-form of (modified) U ı , for τ i = i = w • i , were introduced in [BW18b].2.2. New ı Divided powers for U ı . The Q ( q )-algebra U ı = U ı ς , for ς = ( ς i ) i ∈ I ◦ (subject tosome constraints [BK15, BW18b]), can be defined as a subalgebra of U (similar to e U ı as asubalgebra of e U ). In particular, for i ∈ I ◦ with τ i = i , we have B i = F i + ς i T w • ( E i ) K − i ∈ U ı . Alternatively, U ı is related to e U ı by a central reduction: U ı ς = e U ı / (cid:0)e k i − ς i ( i = τ i ) , e k i e k τi − ς i ς τi ( i = τ i ) (cid:1) . Let us specialize to the case τ i = i ∈ I ◦ , which is most relevant to us. Our parameter ς i corresponds to the notation in [BK15] as ς i = − c i s ( i ). The parameters c i , s ( i ) thereinwere not needed separately. Similarly, the notation Z i = − s ( i ) r i ( T w • E i ) in [BK15] is neverneeded separately, and instead c i Z i and r i ( T w • E i ) are all one needs. We have c i Z i = ς i r i ( T w • E i ) . (2.6)By a slight abuse of notation, the ı divided powers in U ı , denoted again by B ( m ) i,p , for p ∈ Z ,are defined in almost the same way as in (2.3)–(2.4), with e k i replaced by ς i : B ( m ) i, ¯1 = 1[ m ] i ! B i Q kj =1 (cid:16) B i − [2 j − i q i ς i r i ( T w • E i ) (cid:17) if m = 2 k + 1 , Q kj =1 (cid:16) B i − [2 j − i q i ς i r i ( T w • E i ) (cid:17) if m = 2 k ;(2.7) B ( m ) i, ¯0 = 1[ m ] i ! B i Q kj =1 (cid:16) B i − [2 j ] i q i ς i r i ( T w • E i ) (cid:17) if m = 2 k + 1 , Q kj =1 (cid:16) B i − [2 j − i q i ς i r i ( T w • E i ) (cid:17) if m = 2 k. (2.8)2.3. Universality.
For i, j ∈ I , n >
0, denote(2.9) f − i,j ; n,m,e := X r + s = m ( − r q er (1 − na ij − m ) i F ( r ) i F ( n ) j F ( s ) i . The following Serre-Lusztig relations hold in U (cf. [Lus94]): f − i,j ; n,m,e = 0 , for m ≥ − na ij . For τ i = i ∈ I ◦ and i = j ∈ I ◦ , we denote(2.10) S i,j ; n ( B i , B j ) := X r + s =1 − na ij ( − r (cid:20) − na ij r (cid:21) i B ri B nj B si . We formulate the following universality result.
Proposition 2.2 (Letzter, Kolb, De Clercq, ...) . For τ i = i ∈ I ◦ and i = j ∈ I ◦ , we havean identity (Serre-Lusztig type relation) in U ı of the form S i,j ; n ( B i , B j ) = C i,j ; n ( B i , B j ) , (2.11) ERRE-LUSZTIG RELATIONS FOR ı QUANTUM GROUPS II 5 where C i,j ; n ( B i , B j ) is a (non-commutative) polynomials in B i , B j of the form C i,j ; n ( B i , B j ) = X r + s ≤− − na ij ̺ ( i,j,a ij ) r,s | n (cid:0) ς i r i ( T w • E i ) (cid:1) − naij − r − s B ri B nj B si , (2.12) for some universal Laurent polynomials ̺ ( i,j,a ij ) r,s | n ∈ Z [ q, q − ] . (The identity (2.11) remains valid in e U ı when ς i in C i,j ; n ( B i , B j ) above is replaced by e k i .) Sketch of a proof.
In case of ( n = 1 , m = 1 − a ij ), the universality result has been (somewhatimplicitly) known in [Le02, Ko14] and made explicit in [DeC19]. Actually, very careful andtedious work was devoted in [DeC19] to describe explicitly these universal polynomials ̺ ( i,j,a ij ) r,s | n in C i,j ; n ; see [DeC19, Theorem 4.7]. Observing that c i and Z i loc. cit. always appear withthe same power, we can group c i Z i together and replace it by ς i r i ( T w • E i ) as in (2.12).The main method used in these works is the so-called projection techniques which goesback to Letzter, and this leads to [Ko14, Proposition 5.16, Corollary 5.17] which describe theSerre type relations in U ı as F ij ( B i , B j ) = C ij ;1 , where F ij ( B i , B j ) is exactly S i,j ;1 ( B i , B j )in our notation above. If we go through the same arguments loc. cit. , we find the resultsin [Ko14, Proposition 5.16, Corollary 5.17] remain valid for S i,j ; n ( B i , B j ) (with λ ij thereinreplaced by nλ ij accordingly), for n ≥
1. Then the proposition follows formally as in [DeC19].As we do not need to know the universal polynomials ̺ ( i,j,a ij ) r,s | n explicitly (which are perhaps nottoo useful in practice), the painstaking work as in [DeC19] of writing down these polynomialsis not required here. (cid:3) The Serre-Lusztig relations in ı quantum groups The Serre-Lusztig relations of minimal degree.
We consider ı quantum groups U ı of arbitrary Kac-Moody type, where I • = ∅ is allowed. Theorem 3.1 (Serre-Lusztig relations of minimal degree) . For any i = j ∈ I ◦ such that τ i = i , n ∈ Z ≥ , and t ∈ Z , the following identities hold in e U ı : X r + s =1 − na ij ( − r B ( r ) i,p B nj B ( s ) i,p + na ij = 0 , (3.1) X r + s =1 − na ij ( − r B ( r ) i,p B ( n ) j,t B ( s ) i,p + na ij = 0 , for n ≥ . (3.2)We do not recall the precise formulas for the ı divided powers B ( n ) j,t in 3 cases, and we referto [BW18b, (5.12)] and [CLW20, (5.5)] for details.For n = 1, the identity (3.1) reduces to the ı Serre relation (1.3) in U ı , which is alreadynew for arbitrary e U ı . (In case w • i = i and thus r i ( T w • E i ) = 1, the relations in Theorem 3.1reduces to [CLW20, Theorems A], and the ı Serre relation (1.3) was obtained in [CLW18].)
Proof of Theorem 3.1.
As explained in [CLW20, Introduction], (3.2) follows from (3.1) by[CLW20, Proposition 3.2]. Hence it suffices to prove (3.1).By Proposition 2.2, we have a Serre-Lusztig type relation (for i = j ∈ I ◦ ) of the form(2.11), i.e., S i,j ; n ( B i , B j ) = C i,j ; n ( B i , B j ). XINHONG CHEN, MING LU, AND WEIQIANG WANG
The above discussion remains valid in the setting of quasi-split ı quantum groups where r i ( T w • E i ) = 1, where we already have a Serre-Lusztig relation (for i = j ∈ I ◦ ) of the form(3.1), where ς i r i ( T w • E i ) reduces to ς i in the definition of B ( r ) i,p ; see [CLW18, (3.9)]. Then wehave the following expansion in terms of (non-commutative) monomials in B i , B j , where itis understood that r i ( T w • E i ) = 1 in the ı divided powers and in C ij ( B i , B j ): − a ij X r =0 ( − r B ( r ) i,a ij + p i B j B (1 − a ij − r ) i,p i | r i ( T w • E i )=1 = [1 − a ij ] i ! − (cid:16) S ij ( B i , B j ) − C ij ( B i , B j ) | r i ( T w • E i )=1 (cid:17) . Indeed, the universal polynomials e ρ ( i,j,a ij ) m,m ′ are determined from the expansion of the LHSabove; compare [DeC19].The above formula remains valid when replacing the scalar ς i by ς i r i ( T w • E i ) = c i Z i (whichcan be regarded as a commuting variable by (2.2)), that is, − a ij X r =0 ( − r B ( r ) i,a ij + p i B j B (1 − a ij − r ) i,p i = [1 − a ij ] i ! − (cid:0) S ij ( B i , B j ) − C ij ( B i , B j ) (cid:1) . Hence, the identity (3.1) follows from (2.11). (cid:3)
Remark . It is instructive to compare the ı Serre relation in a canonical form (1.3) (asin [CLW18]) to a complicated formulation in [DeC19, Theorem 4.7]. They coincide up to ascalar multiple of [1 − a ij ] i !. Remark . When the parameter ς i satisfies the Condition [BW18b, (3.7)] (which goes backto [BK15] in some form), it follows from [BW18b, (5.10)] (or [BK15, Theorem 3.11(2)]) that q i ς i r i ( T w • E i ) is bar invariant. Hence the relation (1.3) is manifestly bar invariant in this case.Such a bar invariance was also observed in [DeC19] based on the explicit formulas therein. Remark . Combining the relations obtained by Letzter, Kolb and Balagovic (cf., e.g.,[Le02, Ko14, BK15]) and (1.3) in this paper, all the Serre type relations between B i and B j for ı quantum groups e U ı (or U ı ) of arbitrary Kac-Moody type have been formulated in clean and closed formulas, except in the case when τ ( i ) = i ∈ I ◦ and j ∈ I • ; examples in thisexceptional case can be found in [BK15]. Example 3.5.
Let j = i ∈ I ◦ with τ i = i . With the help of (2.6) , the formula (1.3) specializes to (1) B i B j − [2] i B i B j B i + B j B i = q i c i Z i B j , for a ij = − , (2) B i B j − [3] i B i B j B i + [3] i B i B j B i − B j B i = [2] i q i c i Z i ( B i B j − B j B i ) , for a ij = − .The above two formulas, together with a formula for a ij = − , were earlier obtained in [BK15, Theorem 3.7(2)] by rather involved computations. Definition of e y i,j ; n,m,p,t,e and e y ′ i,j ; n,m,p,t,e . Let i = j ∈ I ◦ be such that τ i = i . For m ∈ Z , n ∈ Z ≥ , e = ± p, t ∈ Z , we define elements e y i,j ; n,m,p,t,e and e y ′ i,j ; n,m,p,t,e in e U ı below, depending on the parity of m − na ij . (They are simply modified from those in thesame notations in [CLW20], with a substitution of q i e k i by q i e k i r i ( T w • E i ).)If m − na ij is odd, we let e y i,j ; n,m,p,t,e =(3.3) ERRE-LUSZTIG RELATIONS FOR ı QUANTUM GROUPS II 7 X u ≥ ( q i e k i r i ( T w • E i )) u n X r + s +2 u = m r = p +1 ( − r q − e (( m + na ij )( r + u ) − r ) i (cid:20) m + na ij − u (cid:21) q i B ( r ) i,p B ( n ) j,t B ( s ) i,p + na ij + X r + s +2 u = m r = p ( − r q − e (( m + na ij − r + u )+ r ) i (cid:20) m + na ij − u (cid:21) q i B ( r ) i,p B ( n ) j,t B ( s ) i,p + na ij o ;if m − na ij is even, then we let e y i,j ; n,m,p,t,e =(3.4) X u ≥ ( q i e k i r i ( T w • E i )) u n X r + s +2 u = m r = p +1 ( − r q − e ( m + na ij − r + u ) i (cid:20) m + na ij u (cid:21) q i B ( r ) i,p B ( n ) j,t B ( s ) i,p + na ij + X r + s +2 u = m r = p ( − r q − e ( m + na ij − r + u ) i (cid:20) m + na ij − u (cid:21) q i B ( r ) i,p B ( n ) j,t B ( s ) i,p + na ij o . If m − na ij is odd, we let e y ′ i,j ; n,m,p,t,e =(3.5) X u ≥ ( q i e k i r i ( T w • E i )) u n X r + s +2 u = m r = p +1 ( − r q − e (( m + na ij )( r + u ) − r ) i (cid:20) m + na ij − u (cid:21) q i B ( s ) i,p B ( n ) j,t B ( r ) i,p + na ij + X r + s +2 u = m r = p ( − r q − e (( m + na ij − r + u )+ r ) i (cid:20) m + na ij − u (cid:21) q i B ( s ) i,p B ( n ) j,t B ( r ) i,p + na ij o ;if m − na ij is even, then we let e y ′ i,j ; n,m,p,t,e =(3.6) X u ≥ ( q i e k i r i ( T w • E i )) u n X r + s +2 u = m r = p +1 ( − r q − e ( m + na ij − r + u ) i (cid:20) m + na ij u (cid:21) q i B ( s ) i,p B ( n ) j,t B ( r ) i,p + na ij + X r + s +2 u = m r = p ( − r q − e ( m + na ij − r + u ) i (cid:20) m + na ij − u (cid:21) q i B ( s ) i,p B ( n ) j,t B ( r ) i,p + na ij o . Serre-Lusztig relations in e U ı . The Serre-Lusztig relations as formulated in [CLW20,Theorems B, C, D] for quasi-split ı quantum groups (upon a substitution of e k i by e k i r i ( T w • E i )in e U ı or ς i r i ( T w • E i ) in U ı ) remain valid for arbitrary ı quantum groups; see Theorems 3.6,3.7 and 3.8 below. The same proofs loc. cit. go through verbatim in the current setting andwill not be reproduced here. Theorem 3.6.
For any i = j ∈ I ◦ such that τ i = i , n, u ∈ Z ≥ , and t ∈ Z , the followingidentities hold in e U ı (or in U ı ): X r + s =1 − na ij +2 u ( − r B ( r ) i,p B nj B ( s ) i,p + na ij = 0 , (3.7) XINHONG CHEN, MING LU, AND WEIQIANG WANG X r + s =1 − na ij +2 u ( − r B ( r ) i,p B ( n ) j,t B ( s ) i,p + na ij = 0 . (3.8) Theorem 3.7.
For i = j ∈ I ◦ such that τ i = i , p, t ∈ Z , n ≥ , and e = ± , the followingidentity holds in e U ı : q − e (2 m + na ij ) i B i e y i,j ; n,m,p,t,e − e y i,j ; n,m,p,t,e B i (3.9) = − [ m + 1] i e y i,j ; n,m +1 ,p,t,e + [ m + na ij − i q − e (2 m + na ij − i e k i r i ( T w • E i ) e y i,j ; n,m − ,p,t,e . Theorem 3.8 (Serre-Lusztig relations) . Let i = j ∈ I ◦ such that τ i = i , p, t ∈ Z , n ≥ ,and e = ± . Then, for m < and m > − na ij , the following identities hold in e U ı : e y i,j ; n,m,p,t,e = 0 , e y ′ i,j ; n,m,p,t,e = 0 . (3.10) Remark . Theorems 3.7 and 3.8 hold if we replace B ( n ) j,t by B nj throughout the definitionsof e y i,j ; n,m,p,t,e and e y ′ i,j ; n,m,p,t,e in (3.3)–(3.6). Theorems 3.7 and 3.8 remain valid over U ı = U ı ς ,once we replace e k i by ς i in the definition of e y i,j ; n,m,p,t,e . References [BK15] M. Balagovic and S. Kolb,
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