Cellularity of the p-Canonical Basis for Symmetric Groups
aa r X i v : . [ m a t h . R T ] S e p Cellularity of the p -Canonical Basis forSymmetric Groups Lars Thorge Jensen
Abstract
For symmetric groups we show that the p -canonical basis can be ex-tended to a cell datum for the Iwahori-Hecke algebra H and that thetwo-sided p -cell preorder coincides with the Kazhdan-Lusztig two-sidedcell preorder. Moreover, we show that left (or right) p -cells inside thesame two-sided p -cell for Hecke algebras of finite crystallographic Coxetersystems are incomparable (Property A). Contents p -canonical Basis and p -Cells . . . . . . . . . . . . . . . . . . 52.5 The Perron-Frobenius Theorem . . . . . . . . . . . . . . . . . . . 6 p -Families and p -Special Modules 74 Left p -Cells Within the Same Two-Sided p -Cell 105 p -Cells for Symmetric Groups 16 p -Cell Preorder and the Dominance Order . . . . . . . 165.2 Cellularity of the p -Canonical Basis . . . . . . . . . . . . . . . . . 20 The Hecke algebra of a crystallographic Coxeter system admits several geometricor algebraic categorifications (see [KL79; EW16]). The canonical bases arisingfrom these categorifications coincide with the famous Kazhdan-Lusztig basis (see[KL80; EW14]) in the characteristic 0 setting and give rise to the p -canonical or p -Kazhdan-Lusztig basis in the characteristic p > p -canonical basis of the Hecke algebrawas initiated in [Jen20] and continued in [JP19]. In this paper, we apply thePerron-Frobenius theorem to p -cells and to tackle several questions about the1 INTRODUCTION p -cell preorders. One of the most important features of the Kazhdan-Lusztig cell preorders is Property A which states that left (or right) Kazhdan-Lusztig cells in the same two-sided Kazhdan-Lusztig cell are incomparable withrespect to the left (or right) cell preorder. We will prove Property A of p -cells forfinite crystallographic Coxeter groups in this paper following ideas of [KM16].Along the way, we introduce p -special modules and p -families which aregeneralizations of Lusztig’s special modules and families of irreducible represen-tations of a Weyl group to the p -canonical basis. These concepts were originallyintroduced by Lusztig in [Lus79b; Lus79a; Lus82] and have played an importantrole in determining the complex irreducible characters of finite reductive groups(via character sheaves). The connections of p -special modules and p -families tothe representation theory of finite reductive groups are completely unclear tothe author and thus merit further study.In the last part of the paper, we study consequences of Property A for p -cellsof symmetric groups. One of the main results of [Jen20] is the characterization of p -cells for symmetric groups in terms of the Robinson-Schensted correspondencewhich gives a bijection w ( P ( w ) , Q ( w )) between the symmetric group S n andpairs of standard tableaux of the same shape with n boxes. As a consequence p -cells and Kazhdan-Lusztig cells coincide for all primes p : The two-sided cellof w ∈ S n is given by the set of elements in S n whose Q -symbols have thesame shape as Q ( w ). In other words, two-sided cells of S n are in bijection withpartitions of n .Even though the cells coincide, it was an open question to relate the two-sided p -cell preorder and the Kazhdan-Lusztig two-sided cell preorder on thelevel of two-sided cells. As a special case of the Lusztig-Vogan bijection (whichwas proven in [Bez09]) the Robinson-Schensted correspondence gives an order-preserving bijection between the set of two-sided cells equipped with the Kazhdan-Lusztig two-sided cell preorder and the set of partitions equipped with the domi-nance order. We extend this result to the two-sided p -cell preorder showing thatthe two-sided p -cell preorder and the Kazhdan-Lusztig two-sided cell preordercoincide for all primes p .When Graham and Lehrer introduced cellular algebras in [GL96], the Heckealgebra of a symmetric group together with the Kazhdan-Lusztig basis was oneof the motivating examples. Finally, we show that the p -canonical basis can alsobe extended to a cell datum for the Hecke algebra of a symmetric group. Let’srecall the main result of [LM20, Theorem 1.1]: Theorem 1.1.
Let x, y ∈ S n with x y . Then there exist N > n and v, w ∈ S N with v w such that (i) v and w belong to the same right cell, (ii) the singularity of X w at v is smoothly equivalent to the singularity of X y at x . As explained in [LM20, §4.3] this result allows to embed any example from[Wil17] in which the Kazhdan-Lusztig and the p -canonical basis differ into thesame cell in some larger symmetric group and thus produces many exampleswhere the p -canonical basis gives interesting basis of Specht modules. Thisadds to the interest in this newly defined cell datum for the Hecke algebra of asymmetric group. BACKGROUND Section 2
We introduce notation for crystallographic Coxeter systems andtheir Hecke algebras. Then we recall important results about the diagram-matic category of Soergel bimodules and the p -canonical basis. Finally,we remind the reader of the Perron-Frobenius Theorem. Section 3
We apply the Perron-Frobenius Theorem to p -cells to introduce p -special modules and p -families and to prove some of their elementary prop-erties. Section 4
We prove Property A for finite Weyl groups, showing that left p -cellswithin the same two-sided p -cell are incomparable with respect to the left p -cell preorder. Section 5
This section deals with some consequences for p -cells of symmetricgroups. First, we recall the characterization of p -cells in terms of theRobinson-Schensted correspondence. Then we show that the two-sided p -cell preorder corresponds to the dominance order on partitions. Finally,we prove that the p -canonical basis can be extended to a cellular datumfor the Hecke algebra. Since this project goes back to a question by Peter McNamara at the conferenceon “New Connections in Representation Theory” in Mooloolaba, Australia, inFebruary 2020, I would like to thank him for asking me the question and laterencouraging me to write up the answer. Moreover, I am grateful to the HenriPoincaré Institute in Paris for the good working conditions and to Simon Richeand the ERC project ModRed for the financial support for my research visitin Sydney. I would also like to thank Leonardo Patimo and Shotaro Makisumifor helpful discussions. The author has received funding from the EuropeanResearch Council (ERC) under the European Union’s Horizon 2020 researchand innovation programme (grant agreement No 677147).
Let S be a finite set and ( m s,t ) s,t ∈ S be a matrix with entries in N ∪ {∞} suchthat m s,s = 1 and m s,t = m t,s > s = t ∈ S . Denote by W thegroup generated by S subject to the relations ( st ) m s,t = 1 for s, t ∈ S with m s,t < ∞ . We say that ( W, S ) is a
Coxeter system and W is a Coxeter group .The Coxeter group W comes equipped with a length function l : W → N andthe Bruhat order (see [Hum90] for more details). A Coxeter system ( W, S ) iscalled crystallographic if m s,t ∈ { , , , , ∞} for all s = t ∈ S . We denote theidentity of W by e . For w ∈ W we define its left descent set via L ( w ) := { s ∈ S | l ( sw ) < l ( w ) } .The right descent set of w is given by R ( w ) := L ( w − ). BACKGROUND A = ( a i,j ) i,j ∈ J (see [Tit89,§1.1]). Let ( J, X, { α i : i ∈ J } , { α ∨ i : i ∈ J } ) be an associated Kac-Moody rootdatum (see [Tit89, §1.2] for the definition). Then X is a finitely generated freeabelian group, and for i ∈ J we have elements α i and α ∨ i of X and X ∨ =Hom Z ( X, Z ) respectively that satisfy a i,j = α ∨ i ( α j ) for all i, j ∈ J .To A we associate a crystallographic Coxeter system ( W, S ) as follows:Choose a set of simple reflections S of cardinality | J | and fix a bijection S ∼ → J , s i s . For s = t ∈ S we define m s,t to be 2, 3, 4, 6, or ∞ if a i s ,i t a i t ,i s is 0, 1,2, 3, or > W is a finite Weyl group.Fix a commutative ring k . k V := X ∨ ⊗ Z k yields a balanced, potentiallynon-faithful realization of the Coxeter system over k . Set k V ∗ := Hom k ( k V, k )and note that k V ∗ is isomorphic to X ⊗ Z k . A realization obtained in this wayis called a Cartan realization (see [AMRW17a, §10.1]). Throughout, we willassume our realization to satisfy:
Assumption 2.1 (Demazure Surjectivity) . The maps α s : k V → k and α ∨ s : k V ∗ → k are surjective for all s ∈ S . This is automatically satisfied if 2 is invertible in k or if the Coxeter system( W, S ) is of simply-laced type and of rank | S | > R = S ( k V ∗ ) the symmetric algebra of k V ∗ over k and view it asa graded ring with k V ∗ in degree 2. Given a graded R -bimodule B = L i ∈ Z B i ,we denote by B (1) the shifted bimodule with B (1) i = B i +1 . The Hecke algebra is the free Z [ v, v − ]-algebra with { H w | w ∈ W } as basis,called the standard basis , and multiplication determined by: H s = ( v − − v ) H s + 1 for all s ∈ S , H x H y = H xy if l ( x ) + l ( y ) = l ( xy ).There is a unique Z -linear involution ( − ) on H satisfying v = v − and H x = H − x − . The Kazhdan-Lusztig basis element H x is the unique element in H x + P y Theorem 2.2 (Properties of H ) . Let k be a complete, local, integral domain (e.g. a field or the p -adic integers Z p ). (i) H is a Krull-Schmidt category. (ii) For all w ∈ W there exists a unique indecomposable object k B w ∈ H which is a direct summand of w for any reduced expression w of w andwhich is not isomorphic to a grading shift of any direct summand of anyexpression v for v < w . In particular, the object k B w does not depend upto isomorphism on the reduced expression w of w . (iii) The set { k B w | w ∈ W } gives a complete set of representatives of theisomorphism classes of indecomposable objects in H up to grading shift. (iv) There exists a unique isomorphism of Z [ v, v − ] -algebras ch : [ H ] −→ H sending [ k B s ] to H s for all s ∈ S , where [ H ] denotes the split Grothendieckgroup of H . ( We view [ H ] as a Z [ v, v − ] -algebra as follows: the monoidalstructure on H induces a unital, associative multiplication and v acts via v [ B ] := [ B (1)] for an object B of H . )It should be noted that we do not have a diagrammatic presentation of H as determining the idempotents in BS is usually extremely difficult. p -canonical Basis and p -Cells In this section, we recall the definition of the p -canonical basis and its elementaryproperties (see [JW17; Jen20]). Let k be a field of characteristic p > 0. Notethat the p -canonical basis depends on p , but not on the explicit choice of k . Definition 2.3. Define p H w = ch([ k B w ]) for all w ∈ W where ch : [ H ] ∼ = −→ H is the isomorphism of Z [ v, v − ]-algebras introduced earlier.We will frequently use the following elementary properties of the p -canonicalbasis which can be found in [JW17, Proposition 4.2] unless stated otherwise: Proposition 2.4. For all x, y ∈ W we have: (i) p H x = p H x , i.e. p H x is self-dual, (ii) ι ( p H x ) = p H x − and thus in particular p m y,x = p m y − ,x − as well as p h y,x = p h y − ,x − , (iii) p H x p H y = P z ∈ W p µ zx,y p H z with self-dual p µ zx,y ∈ Z > [ v, v − ] , (iv) p H x = H x for p = 0 (see [EW14]) and p ≫ (i.e. there are only finitelymany primes for which p H x = H x ). BACKGROUND p -cells (see [Jen20, §3.1] for more details)which is an obvious generalization of a notion introduced by Kazhdan-Lusztigin [KL79]: Definition 2.5. For h ∈ H we say that p H w appears with non-zero coefficientin h if the coefficient of p H w is non-zero when expressing h in the p -canonicalbasis.Define a preorder p R (resp. p L ) on W as follows: x p R y (resp x p L y ) if andonly if p H x appears with non-zero coefficient in p H y h (resp. h p H y ) for some h ∈ H . Define p LR to be the preorder generated by p R and p L , in other wordswe have: x p LR y if and only if p H x appears with non-zero coefficient in h p H y h ′ for some h, h ′ ∈ H .The left, right, or two-sided p -cells are the equivalence classes with respectto the corresponding preorders respectively.Let R be a fixed commutative ring with unit and let A be an R -algebra withfixed R -basis indexed by B . For a subset J ⊆ B we denote by A ( J ) the R -spanof the basis elements indexed by elements in J . Let C be a left p -cell in W . Tosimplify notation, we write p L C := { x ∈ W | x p L y for some y ∈ C } p < L C := { x ∈ W | x p L y for some y ∈ C and y / ∈ C } Unless stated otherwise, we will consider throughout the p -canonical basis of H (and its extension of scalars to C ). Recall that C gives rise to a cell module H ( C ) = H ( p L C ) / H ( p < L C )which is a left module for H (similarly for right or two-sided p -cells). The p -canonical basis elements indexed by elements in C gives rise to a basis of H ( C ). Our main technical tool will be the Perron-Frobenius theorem (see [Per07; Fro08;Fro09]) which was published over a 100 years ago. A more modern expositioncan be found in [Gan59, Vol. 2, Chapter XIII] or in [Hup90, Kapitel IV]. Theorem 2.6 (Perron-Frobenius) . Let M ∈ Mat k × k ( R > ) . Then there exists λ ∈ R > , called the Perron-Frobenius eigenvalue of M , such that the followingstatements holds: (i) λ is an eigenvalue of M . (ii) Any other eigenvalue µ ∈ C of M satisfies | µ | < λ , so λ gives the spectralradius of M . (iii) The eigenvalue λ has algebraic multiplicity . P -FAMILIES AND P -SPECIAL MODULES There exists v ∈ R k> such that M v = λv . There exists also ˆ v ∈ R k> suchthat ˆ v T M = λ ˆ v T . (v) Any w ∈ R k > which is an eigenvector for M is a scalar multiple of v andsimilarly for ˆ v . (vi) If v and ˆ v are normalized such that ˆ v T v = (1) , then lim n →∞ M n λ n = v ˆ v T . p -Families and p -Special Modules In this section, we will apply some results of [KM16] to the p -canonical basis ofthe complex group ring of a finite Weyl group. Thus we assume that ( W, S ) isa finite Weyl group throughout the section. For the sake of completeness, wegive all the proofs.Denote by C W = C ⊗ Z [ v,v − ] H the scalar extension of H to C where wespecialize v to 1. For the rest of the paper fix a set of coefficients c = { c w } w ∈ W with c w ∈ R > . c determines a Perron-Frobenius element p b c = X w ∈ W c w ⊗ p H w ∈ C W .Let C be a left p -cell in W . We will denote by C W ( C ) = C ⊗ Z [ v,v − ] H ( C )the extension of scalars of the corresponding left cell module H ( C ) to C . Asan immediate consequence of Theorem 2.6 we get in this setting (see [KM16,Corollary 3]): Corollary 3.1. The left (resp. right) action of p b c on C W ( C ) gives a Perron-Frobenius eigenvalue p a c ( C ) . There is up to isomorphism a unique irreducible C W -module L C, c occuring as composition factor of the cell module C W ( C ) suchthat p a c ( C ) is afforded by the action of p b c on L C, c . Moreover, [ C W ( C ) : L C, c ] is equal to . The following result shows that we can simplify our notation (see [KM16,Theorem 5]): Theorem 3.2. L C, c does up to isomorphism not depend on the choice of c .Proof. For simplicity, denote by n = | W | the cardinality of W . Consider themap L C, − : R n> → Irr( C W ) which sends d to L C, d . Equipping Irr( C W ) withthe discrete topology, we claim that this map is continuous. Since R n> is con-nected, any continuous function to a discrete set has to be constant. Therefore,the claim implies the theorem.To prove the claim, we will show that the preimage X L ⊆ R n> of any L ∈ Irr( C W ) under this map is closed. Assume X L is non-empty and let d i ∈ R n> ∩ X L be a sequence that converges to d ∈ R n> . Let L , L , . . . , L k = L be the list of simple subquotients of C W ( C ). Denote by M d i the linear operatoron C W ( C ) the element p b d i ∈ C W induces. As d i ∈ X L , we have for all i ∈ N by Theorem 2.6: p a d i ( C ) = max {| µ | | µ ∈ Spec( M d i | L ) } P -FAMILIES AND P -SPECIAL MODULES > sup j For any other left p -cell C ′ which belongs to the same two-sided p -cell as C , we have L C ∼ = L C ′ and p a c ( C ) = p a c ( C ′ ) .Proof. Denote by J the two-sided cell that contains C and C ′ . We may assumethat C ′ is maximal with respect to p L in J .First, we will construct a non-zero homomorphism ϕ : C W ( C ) −→ C W ( C ′ )of C W -modules. For any u ∈ C and v ∈ C ′ there exist h ′ , h ∈ H such that p H v occurs with non-zero coefficient in h ′ p H u h by the definition of two-sided p -cells.It follows that p H u h intersects the set { x ∈ W | x p > L v } ∩ J non-trivially. Byour assumption of maximality the intersection { x ∈ W | x p > L v } ∩ J coincideswith C ′ . Thus right multiplication by h and projection onto C W ( C ′ ) definesthe non-zero homomorphism ϕ . Observe that ϕ sends any linear combinationof the p -canonical basis of C W ( C ) with strictly positive coefficients to a non-zero linear combination of the p -canonical basis of C W ( C ′ ) with non-negativecoefficients.By Theorem 2.6 (iv) there exists an eigenvector v of p b c on C W ( C ) witheigenvalue p a c ( C ) that is a linear combination of the p-canonical basis withstrictly positive coefficients. It follows that ϕ ( v ) is non-zero and an eigenvectorof p b c in C W ( C ′ ) with eigenvalue p a c ( C ). Moreover, ϕ ( v ) is a linear combinationof the p -canonical basis of C W ( C ′ ) with non-negative coefficients. Therefore, thecorresponding eigenvalue is the Perron-Frobenius eigenvalue of p b c on C W ( C ′ )by Theorem 2.6 (v). This implies p a c ( C ) = p a c ( C ′ ). As v ∈ L C , the simplesubquotient L C is not annihilated by ϕ . Schur’s lemma implies that ϕ inducesan isomorphism L C ∼ = L C ′ .Kildetoft and Mazorchuk prove the following interesting results as well (see[KM16, Proposition 13]): Proposition 3.4. (i) The number of left p -cells in the two-sided p -cell of C is given by dim L C . (ii) The two sided p -cells induce a partition of the irreducible representationsof W as follows: Irr( W ) = [ J ⊆ W two-sided p -cell { L ∈ Irr( W ) | [ C W ( J ) , L ] = 0 } P -FAMILIES AND P -SPECIAL MODULES In particular, any simple subquotient of C W ( C ) different from L C is notisomorphic to L C ′ for any other left p -cell C ′ .Proof. Fix a total order J , J , . . . , J k of the two-sided p -cells of W such that i < j implies J i p > J j . For 0 i k denote by I i the linear span of 1 ⊗ p H x for x ∈ J s with s i . Then0 = I ⊂ I ⊂ I ⊂ · · · ⊂ I k = C W (1)is a filtration of C W by two-sided ideals.Since C W is a semisimple algebra, each simple C W -module L occurs in theleft regular representation with multiplicity dim( L ). As (1) is a filtration bytwo-sided ideals, there exists an index 1 i k such that L appears withmultiplicity dim( L ) in I i /I i − . On the other hand, I i /I i − is isomorphic to thedirect sum of all the cell modules C W ( C ) for C a left p -cell in J i . This impliespart (ii) of the Proposition. Finally, observe that L occurs exactly once in eachof these left cell modules. This proves part (i) and finishes the proof of theProposition.We extend p a c to a function W → R ≥ mapping w contained in the left p -cell C to p a c ( C ). Using the characterization of the Perron-Frobenius eigenvalue asthe largest real eigenvalue, we get immediately from Theorem 3.3: Corollary 3.5. p a c is constant on two-sided p -cells. Moreover, we would like to prove: Conjecture 3.6. For x, y ∈ W we have: x p y ⇒ p a c ( x ) > p a c ( y )Our current techniques do not allow us to prove this as we do not under-stand the multiplication of the p -canonical basis well enough. Using Computercalculations we have verified the conjecture for finite Weyl groups of rank Definition 3.7. Let J ⊆ W be a two-sided p -cell and C ⊆ J a left (or right) p -cell. We call L C the p -special module of J and the set { L ∈ Irr( W ) | [ C W ( J ) , L ] = 0 } the p -family of J .In the case of the Kazhdan–Lusztig basis and a left (or right) Kazhdan–Lusztig cell C , the module L C is a special representation in the sense of Lusztig(see [KM16, Proposition 9]). In [Lus84, Theorem 5.25] Lusztig shows that forthe Kazhdan–Lusztig basis our notion of families coincides with the originaldefinition. An alternative proof of this result based on the theory of primitiveideals can be found in [BV82; BV83]. LEFT P -CELLS WITHIN THE SAME TWO-SIDED P -CELL Example . In type B we have for the 2-Kazhdan–Lusztig basis (with thesame conventions as in [JW17]): H sts = H sts + H s H w = H w for all w ∈ W \ { sts } The 2-special modules in this case are: { id } the sign representation { s } C where s (resp. t ) acts via 1 (resp. − W \ { id , s, w } the geometric representation { w } the trivial representationThus the irreducible C W -modules fall into the following 2-families: { sgn } ∪ { sgn s } ∪ { sgn t , geom } ∪ { triv } where sgn s denotes the special module associated to the two-sided 2-cell { s } .The difference to Lusztig’s families is that the family { sgn s , sgn t , geom } splitsup into two 2-families. Moreover, observe that tensoring with the sign represen-tation does not give a permutation of the set of 2-families, whereas this is thecase for Lusztig’s families. p -Cells Within the Same Two-Sided p -Cell In this section, we will prove that left (or right) p -cells within the same two-sided p -cell are incomparable. Again we assume that W is a finite Weyl groupthroughout the section. In order to do so, we will need the definition of anidempotent two-sided cell which reads in our setting as follows: Definition 4.1. Let J be a two-sided p -cell. J is called idempotent if thereexist elements x, y, z ∈ J such that p H x occurs with non-trivial coefficient in p H y p H z .Even though it might not be obvious at first, we get the following result (see[KM16, Proposition 13 (i)]): Lemma 4.2. Each two-sided p -cell in W is idempotent.Proof. Let J be a two-sided p -cell. Suppose J is not idempotent. The set { p H x | x p < J } induces a two-sided ideal in H and thus in the semisimple algebra C W which wewill denote by I J . We consider the ideal I spanned by 1 ⊗ p H x for x ∈ J in thesemisimple quotient C W/I J . As J is not idempotent, I is a non-zero nilpotentideal, contradicting the semisimplicity of the quotient C W/I J . (Recall that theJacobson radical contains all nilpotent ideals.) LEFT P -CELLS WITHIN THE SAME TWO-SIDED P -CELL p -cell C , Kildetoft and Mazorchuk study the set X C of all two-sided p -cells J such that there exists x ∈ J satisfying p H x · H ( C ) = 0 (orequivalently p H x · C W ( C ) = 0). Applying [KM16, Propositions 14 and 17] weget the following properties: Proposition 4.3. (i) The set X C contains a minimum element, called theapex of C and denoted J ( C ) . (ii) For all x p > J ( C ) we have p H x · H ( C ) = 0 . (iii) For any two-sided p -cell I and left p -cell C ′ ⊂ I , we have J ( C ′ ) = I . The following result is the main result of this section. In the setting of theKazhdan–Lusztig basis, this result was originally deduced from properties ofprimitive ideals in enveloping algebras (see [Lus81, §4]). It is a weak form of[Lus03, P9]. Theorem 4.4. If x p L y and x p ∼ y , then x p ∼ L y . We will break the proof of the theorem into several lemmata, following alongthe lines of [KM16, Proposition 18 and Corollary 19]. Let J be a two-sided p -celland C ⊆ J a left p -cell. Lemma 4.5. The following statements hold: (i) Let I J be the Z [ v, v − ] -span in H of all p H w such that w p > J . Thenthe cell module H ( C ) is naturally a H /I J -module. Similarly, C W ( C ) isnaturally a C W/ C ⊗ Z [ v,v − ] I J -module. (ii) Define a c = P w ∈ J c w ⊗ p H w ∈ C W . Let M c ,C be the matrix giving theaction of a c on C W ( C ) . Then M c ,C has positive coefficients.Proof. Observe that I J is a two-sided ideal in H . Then the first part followsimmediately from the definition of the apex of C and the fact that the apex of C is J (see Proposition 4.3 (iii)). It remains to prove the second part.First, we claim that all columns of M c ,C are non-zero. Suppose that M c ,C has a zero column indexed by y ∈ C . Since the structure coefficients of the p -canonical basis are Laurent polynomials with non-negative coefficients (seeProposition 2.4 (iii)), this implies p H x p H y ∈ H ( p < L C )for all x ∈ J . Denote by I the Z [ v, v − ]-span of all p H x in H /I J for x ∈ J . Itfollows that I · p H y ⊂ H ( p < L C ).Observe that I is a two-sided ideal in H /I J , so that we get: I · ( H /I J ) · p H y ⊂ H ( p < L C ) LEFT P -CELLS WITHIN THE SAME TWO-SIDED P -CELL p H y generates the whole cellmodule H ( C ) under the action of H /I J . Therefore we must have I · H ( C ) ⊂ H ( p < L C )contradicting the fact that the apex of C is J (see Proposition 4.3 (iii)).Next, we claim that all entries in every column of M c ,C are non-zero. Con-sider the column corresponding to y ∈ C . Let X ⊆ C be the set of all x ∈ C such that the basis element p H x appears with a non-zero coefficient in a c p H y in H ( C ). Above we have shown that X is non-empty. Observe that X containsall x ∈ C such that the basis element p H x appears with a non-zero coefficientin I · p H y in H ( C ). Since I is a two-sided ideal in H /I J , it thus follows thatthe Z [ v, v − ]-span of p H x for x ∈ X in H ( C ) is invariant under the action of H /I J . By the transitivity of the cell module, we see that X = C . Therefore, allentries in M c ,C are positive, which concludes the proof.We will continue with the notation of the previous lemma. From Lemma 4.5(ii) it follows that we can apply the Perron-Frobenius theorem to M c ,C . Let λ denote the Perron-Frobenius eigenvalue of M c ,C . From Theorem 2.6 (vi) wededuce that the matrix M C := lim m →∞ M m c ,C λ m is positive and satisfies M C = M C . Observe that M C is the projector to theone-dimensional λ -eigenspace of M c ,C and thus called the Perron-Frobenius pro-jector . Theorem 4.6. We can define an element e J ∈ C W/ C ⊗ Z [ v,v − ] I J with thefollowing properties: (i) e J = e J . (ii) e J can be written as a linear combination of ⊗ p H x for x ∈ J with positivereal coefficients. (iii) e J acts on C W ( C ) via M C . (iv) Let C , C , . . . , C k be the left p -cells in J . In any total order of the p -canonical basis preserving elements of the same left p -cell as adjacent el-ements, we have that e J acts on C W ( J ) via M C . . . M C . . . ...... . . . . . . . . . M C k .Proof. First, consider for m > a m c λ m = X x ∈ J d x,m p H x ∈ C W/ C ⊗ Z [ v,v − ] I J LEFT P -CELLS WITHIN THE SAME TWO-SIDED P -CELL d x,m ∈ R > . Since for x ∈ J the matrix giving the action of p H x on C W ( C ) is non-zero by Proposition 4.3 (ii) and (iii) and has non-negative coef-ficients, the sequence ( d x,m ) m > is bounded for all x ∈ J and thus contains aconvergent subsequence.We have the following identity: a m +1 c λ m +1 = X x ∈ J d x,m +1 p H x = a c λ · a m c λ m = a c λ · X y ∈ J d y,m p H y = X x,y ∈ J N x,y λ d y,m p H x where N is the matrix giving the action of a c on the two-sided p -cell module C W ( J ). This implies for all x ∈ J : d x,m +1 = X y ∈ J N x,y λ d y,m = · · · = X y ∈ J ( N m ) x,y λ m d y, (2)Let C , C , . . . , C k be an ordering of the left p -cells in J such that C i p L C j implies i j . We choose a total order of the p -canonical basis in J refining thetotal order of the left p -cells in J . Then observe that N has the following blockupper-triangular form N = M c ,C ∗ . . . ∗ M c ,C . . . ...... . . . . . . ∗ . . . M c ,C k where M c ,C l is the matrix giving the action of a c on the left p -cell module C W ( C l ) for 1 l k . Recall that the spectrum of N is the multiset unionof the spectra of the diagonal block matrices M c ,C l for 1 l k . Therefore,Theorem 2.6 implies that λ is an eigenvalue with multiplicity k for N and thatany other eigenvalue µ = λ satisfies | µ | < λ .Above we have shown that the sequence ( d x,m ) m > has a convergent sub-sequence for all x ∈ J . This implies that lim m →∞ N m λ m exists and consequentlythat for all x ∈ J the whole sequence ( d x,m ) m > converges to some d x ∈ R > .In addition, it follows that the geometric and algebraic multiplicity of λ for N coincide (see [Mey00, (7.10.33)]).Next, we define e J := X x ∈ J d x p H x .It follows immediately that e J = e J as e J is the projector to the λ -eigenspaceof N and that e J acts via M C on C W ( C ). This concludes the proof of (i) and(iii). LEFT P -CELLS WITHIN THE SAME TWO-SIDED P -CELL e J on the two-sided p -cell module C W ( J ) in the p -canonical basis is given by the following matrix: N J := lim m →∞ N m λ m = M C ∗ . . . ∗ M C . . . ...... . . . . . . ∗ . . . M C k which is block upper-triangular with positive, idempotent matrices on the di-agonal (by the considerations preceding the proposition). Since N J is a non-negative idempotent matrix, we may apply [Flo69, Theorem 2] to get that alloff-diagonal blocks are 0. (Actually, the easy special case where neither a rownor a column of the non-negative idempotent matrix is zero suffices.) Therefore, N J is the direct sum of the positive, idempotent matrices M C i for 1 i k .This completes the proof of part (iv).It remains to prove part (ii). From (2), we deduce that: d x = X y ∈ J N x,y c y, = X l k X y ∈ C l ( M C l ) x,y c y λ > e J projects in each cell module C W ( C l )for 1 l k to the the unique (up to scalar) non-negative eigenvector of M C l by Theorem 2.6 (iv) and (v). Moreover, the image of e J in C W ( J ) gives apositive eigenvector for N J for the eigenvalue 1 which is not unique though as1 is a semisimple eigenvalue of multiplicity k .Recall that the coefficients p µ zx,y for x, y, z ∈ W are the structure coefficientsof the p -canonical basis (see Proposition 2.4 (iii)). The reader should comparethe following corollary with [Lus03, P8] and keep in mind that γ x,y,z is thecoefficient in front of the highest (or lowest) power of v in µ z − x,y . One wouldobtain the third cell equivalence if cyclicity (see [Lus03, P7]) held. Corollary 4.7. Let x, y, z ∈ J . If p µ zx,y is non-zero, then y p ∼ L z and x p ∼ R z .In other words, z lies in the intersection of the right p -cell of x with the left p -cell of y . Moreover, any p H z for z ∈ J occurs with non-trivial coefficient ina product p H x · p H y with x, y ∈ J .Proof. Theorem 4.6 (ii) shows that e J is a positive linear combination of all p -canonical basis elements indexed by elements in J . The block-diagonal formof the matrix N J giving the action of e J on the two-sided p -cell module C W ( J )(see Theorem 4.6 (iv)) demonstrates that p µ zx,y non-zero implies y p ∼ L z .To see that any p H z for z ∈ J occurs with non-trivial coefficient in a product p H x · p H y with x, y ∈ J , recall that all the diagonal blocs in N J have positivecoefficients.Recall that J − is a two-sided p -cell as well. Applying the Z [ v, v − ]-linearanti-involution ι , we obtain p µ zx,y = p µ z − y − ,x − . Combined with the first part ofthe corollary applied to J − we see that p µ zx,y non-zero gives x − p ∼ L z − whichis equivalent to x p ∼ R z . LEFT P -CELLS WITHIN THE SAME TWO-SIDED P -CELL Corollary 4.8. The left p -cells within J are incomparable with respect to theleft p -cell preorder. In other words, we have for x, y ∈ J : x p L y ⇒ x p ∼ L y Proof. Let I be the Z [ v, v − ]-span of all p H z in H /I J for z ∈ J . If x p L y , thenthere exists h ∈ H such that p H x occurs with non-trivial coefficient in h · p H y .From the previous corollary it follows that there are v, w ∈ J such that p µ yv,w is non-zero. Thus, p H x occurs with non-trivial coefficient in h p H v p H w . Since I is a two-sided ideal in H /I J , it follows that the image of h p H v p H w in H /I J lies in I . Rewriting ( h p H v ) p H w in H /I J in terms of the p -canonical basis andapplying the previous corollary shows that x lies in the same left p -cell as w which in turn lies in the same left cell as y . Corollary 4.9. Two-sided p -cells are the smallest subsets that are at the sametime a union of left p -cells and one of right p -cells.Example . Let us illustrate the results of this section in type B con-tinuing Example 3.8. Let us consider the most interesting two-sided 2-cell J = { , , , , } which decomposes into two left 2-cells J = { , , } ∪ { , } and the element a c = P x ∈ J p H x (i.e. all constants are chosen to be 1). Thematrix giving the action of a c on C W ( J ) looks as follows N := where we omitted the 0 entries. N has as Perron-Frobenius eigenvalue λ =6 + 4 √ 2. The eigenvectors to this eigenvalue for multiplication with N on theright are given byˆ v T1 := (cid:16) √ (cid:17) and ˆ v T2 := (cid:16) √ (cid:17) .Similarly, the eigenvectors to this eigenvalue for multiplication with N on theleft are given by v := ˆ v and v := √ . P -CELLS FOR SYMMETRIC GROUPS N J : N J := lim m →∞ N m λ m = √ 24 14 √ 24 12 √ √ 24 14 12 √ √ 22 12 = 14 ( v ˆ v T1 + v ˆ v T2 )For the idempotent e J we get: e J := 2 − √ ⊗ ( p H + p H + p H )+ √ − ⊗ ( p H + p H ) ∈ C W/ C ⊗ Z [ v,v − ] H ( p p -Cells for Symmetric Groups p -Cell Preorder and the Dominance Order The Robinson-Schensted correspondence (see [BB05, §A.3.3] or [Ful97, §4.1])gives a bijection between the symmetric group S n and pairs of standard tableauxof the same shape with n boxes. The row-bumping algorithm gives a way toexplicitly calculate the image ( P ( w ) , Q ( w )) of w ∈ S n under the Robinson-Schensted correspondence.Throughout this section we assume that we used the Cartan matrix in finitetype A n − as input. In this case the Coxeter system ( W, S ) can be identifiedwith the pair ( S n , { s , . . . , s n − } ) consisting of the symmetric group togetherwith the set of simple transpositions. p -Cells for symmetric groups admit a beautiful description in terms of theRobinson-Schensted correspondence (see [Jen20, Theorem 4.33]): Theorem 5.1. For x, y ∈ S n we have: x p ∼ L y ⇔ Q ( x ) = Q ( y ) x p ∼ R y ⇔ P ( x ) = P ( y ) x p ∼ LR y ⇔ Q ( x ) and Q ( y ) have the same shapeIn particular, Kazhdan–Lusztig cells and p -cells of S n coincide. The last result shows that partitions of n correspond to two-sided cells of S n . Recall the definition of the dominance order on the set of partitions of n : Definition 5.2. Let λ = ( λ , λ , . . . ) , µ = ( µ , µ , . . . ) be two partitions of n .We have λ < = µ in the dominance order if and only if k X i =1 λ i k X i =1 µ i for all k > P -CELLS FOR SYMMETRIC GROUPS Proposition 5.3. For two partitions λ , µ of n the following holds: λ µ ⇔ λ T > µ T where λ T denotes the conjugate or transpose partition of λ . We will not explicitly need the following definition, but we give it for thesake of completeness (see [BB05, §A3.7 and §A3.8] for details): Definition 5.4. Given a standard tableau T , the combinatorial algorithm called evacuation proceeds as follows:First, delete the entry 1 and perform a backward slide on the cell thatcontained it. Then repeat this for the entry 2, etc. Finally, the tableau thatrecords in reverse the order in which the cells of T have been vacated is calledthe evacuation of T and denoted by e ( T ).For our next result, we will use the following compatibility of the Robinson-Schensted correspondence with the multiplication of the longest element w ∈ S n (see [BB05, Fact A3.9.1] or [Knu98, Theorem D] for a proof): Theorem 5.5. If x ∈ S n corresponds to ( P, Q ) under the Robinson-Schenstedcorrespondence, then we have under the Robinson-Schensted correspondence that (i) xw corresponds to ( P T , e ( Q ) T ) , (ii) w x corresponds to ( e ( P ) T , Q T ) and (iii) w xw corresponds to ( e ( P ) , e ( Q )) where e ( P ) is the evacuation of P . To simplify notation, we will denote by J λ ⊆ S n the two-sided cell corre-sponding to a partition λ of n . In order to prove the main result, we will relyon the following observation which is motivated by [Wil03, Proposition A.2.1]: Proposition 5.6. The dominance order on partitions of n is generated by theweak Bruhat order under the Robinson-Schensted correspondence.Proof. Let λ, µ be two partitions of n such that λ > µ . We will prove thestatement by induction on the length of a maximal chain between λ and µ . For λ = µ there is nothing to show.For the induction step, suppose λ > µ . It is enough to show that there exists s ∈ S , x ∈ J µ and a partition ν of n such that• xs < x ,• xs ∈ J ν and• λ > ν > µ . P -CELLS FOR SYMMETRIC GROUPS ν from µ by applying a single raising operation. It is awell-known fact that the raising operations generate the dominance order (seefor example [Mac95, (1.16)]).Let i ∈ N be minimal such that λ i > µ i and observe that it is the first indexfor which the parts of λ and µ differ. It follows µ i − = λ i − > λ i > µ i and µ i +1 = 0 as µ and λ are both partitions of n .We will distinguish two cases: 1. Case: µ i +1 > µ i +2 . Define the partition ν as follows: v j := µ i + 1 if j = i , µ i +1 − j = i + 1, µ j otherwise.It follows immediately that ν is a partition of n satisfying λ > ν > µ .Our convention of drawing partitions is that ν i gives the number of boxesin row i . With this convention in mind, divide ν into three parts:• the first i − i -th and ( i + 1)-st row and• the remaining rows.Let T be the standard tableau of shape ν that is column superstandardin each of the three pieces (of course up to shift so that the numbers donot repeat). Suppose that there are k boxes in the first i − i and i + 1 of T look as follows: k + 2 k + 4 · · · k + 2 ν i +1 k + 1 k + 3 · · · k + 2 ν i +1 − k + 2 ν i +1 +1 k + 2 ν i +1 +2 · · · k + ν i +1 + ν i Define a permutation y ∈ S n as follows: Let ( y (1) , y (2) , . . . , y ( n )) be thereading word of T (i.e. the sequence obtained by reading the entries of T from bottom to top, left to right). Then the sequence contains thefollowing piece for part 2:( . . . , k + 2 , k + 4 , . . . , k + 2 ν i +1 , k + 1 , k + 3 , . . . , k + 2 ν i +1 − ,k + 2 ν i +1 + 1 , k + 2 ν i +1 + 2 , . . . , k + ν i +1 + ν i , . . . )Observe that k + ν i +1 + ν i and k + ν i +1 + ν i − s ∈ S be the simple transposition swapping these two positions.It follows ys > y as s introduces a new inversion. We have that ys lies in J µ because when applying the row-bumping algorithm the entry k + ν i + ν i +1 is bumped one row down by k + ν i + ν i +1 − i + 1. This finishes the proof in the first case by setting x := ys . P -CELLS FOR SYMMETRIC GROUPS 2. Case: µ i +1 = µ i +2 . Let m ∈ N be maximal such that µ m = µ i +1 . Definethe partition ν as follows: v j := µ i + 1 if j = i and µ i = µ i +1 , µ i +1 + 1 if j = i + 1 and µ i > µ i +1 , µ m − j = m , µ j otherwise.It follows that ν is a partition of n satisfying λ > ν > µ . Indeed, observethat the smallest index k ∈ N such that P kl =1 λ l = P kl =1 µ l satisfies k > m in order to see λ > ν .Next, observe that ν and µ differ in exactly two adjacent columns l and l + 1. By Proposition 5.3, we have ν T < µ T and µ T is obtained from ν T byraising a box from row l +1 to row l . We will define the permutation y ∈ S n and the simple transposition s ∈ S as before, with the only difference that T will be of shape µ T instead of ν . It follows that ys lies in J ν T andsatisfies ys > y . Define a simple reflection t := w sw . Theorem 5.5shows that x := yw lies in J µ and xt = ysw lies in J ν . Moreover, wehave xt = ysw < yw = x . This finishes the proof of the proposition.The goal of this section is to prove the following result: Theorem 5.7. Let J, J ′ ⊆ S n be two-sided cells that correspond to the partitions λ, λ ′ of n respectively. Then we have the following: J p J ′ ⇔ λ λ ′ In particular, the two-sided p -cell preorder coincides with the Kazhdan–Lusztigtwo-sided cell preorder for all primes p .Proof. ⇒ In this case, we have h, h ′ ∈ H such that for some w ∈ J the p -canonical basis element p H w occurs with non-trivial coefficient in h p H w J ′ h ′ where w J ′ is the longest element in a standard parabolic subgroup contained in J ′ . Since the corresponding Schubert variety is smooth, we have p H w J ′ = H w J ′ .This implies w w J ′ and thus J J ′ .Applying the characterization of the two-sided Kazhdan–Lusztig preorder interms of the dominance order (see [Gec06, Theorem 5.1]), we get λ λ ′ .It is important to note that Geck uses the opposite dominance order because hisconnection to two-sided Kazhdan–Lusztig cells is not based on the Robinson-Schensted correspondence, but on leading matrix coefficients of irreducible rep-resentations of H⊗ Z [ v,v − ] R ( v ). It depends on the parametrization of irreduciblerepresentations of S n in a way that is compatible with induced sign charactersfrom Young subgroups (see [Gec06, Example 3.8]). In his correspondence, heintroduces the transpose of a partition which by Proposition 5.3 inverses thedominance order (compare [Gec06, Corollary 5.6] with Theorem 5.1). ⇐ By induction on the maximal chain between λ and µ it is enough to prove J p < J ′ in the case where λ < λ ′ are adjacent in the dominance order (so that P -CELLS FOR SYMMETRIC GROUPS λ µ λ ′ implies µ = λ or µ = λ ′ ). In the proof of Proposition 5.6 we haveshown that in this case there exists x ∈ J , s ∈ S such that xs < x and xs ∈ J ′ .Due to ( xs ) s = x > xs it follows that p H x occurs with non-trivial coefficient in p H xs H s . This implies x p < R xs and thus J p < J ′ finishing the proof.The last result in the setting of the Kazhdan–Lusztig basis is a special caseof a conjecture made independently by Lusztig in [Lus89, §10.8] and Vogan in[Vog00]. Proofs can be found in [DPS98, p. 2.13.1] and [Gec06, Theorem 5.1]for symmetric groups, in [Shi96] for affine Weyl groups of type e A n or of rank p -cell preorder andthe Kazhdan–Lusztig left (or right) cell preorder. It should also be noted thata charaterization of either of these preorders on standard tableau under theRobinson-Schensted correspondence is not known. p -Canonical Basis First, we will recall the definition of a cellular algebra given in [GL96, Definition1.1]. Let R be a fixed commutative ring with unit and let A be an R -algebra. Definition 5.8. A cell datum for A is a quadruple (Λ , ∗ , M, C ) consisting of:• A finite partially ordered set Λ,• An R -linear anti-involution ∗ of A ,• For every λ ∈ Λ a finite, non-empty set M ( λ ) of indices, and• An injective map C : S λ ∈ Λ M ( λ ) × M ( λ ) → A . If λ ∈ Λ and S, T ∈ M ( λ )write C λS,T = C ( S, T ) ∈ A .and satisfying the following conditions:(i) The image of C gives an R -basis of A .(ii) ( C λS,T ) ∗ = C λT,S .(iii) If λ ∈ Λ and S, T ∈ M ( λ ) then we have for any element a ∈ AaC λS,T = X S ′ ∈ M ( λ ) r a ( S ′ , S ) C λS ′ ,T (mod A ( < λ ))where the coefficients r a ( S ′ , S ) ∈ R are independent of T and where A ( <λ ) denotes the R -submodule of A generated by { C µV,W | µ < λ, V, W ∈ M ( µ ) } .Examples of algebras that can be equipped with a cell datum include matrixrings Mat d × d ( R ), R [ x ] / ( x n ), the Temperley Lieb algebra and many more (see[GL96, §1, §4-6] for the details).For the rest of this section we assume again that we used the Cartan matrixin finite type A n − as input for the Hecke category.The goal of this section is to extend the p -canonical basis to a cell datum of H . The cell datum will be chosen as follows: P -CELLS FOR SYMMETRIC GROUPS 21• Let Λ be the set of partitions of n , equipped with the dominance order.• Let ∗ be the Z [ v, v − ]-linear anti-involution ι of H .• For λ ∈ Λ, let M ( λ ) be the set of standard tableaux of shape λ .• If w ∈ S n corresponds to ( P, Q ) of shape λ under the Robinson-Schenstedcorrespondence, then define C to map ( P, Q ) to p H w . Since the Robinson-Schensted correspondence gives a bijection between S n and the set ofpairs of standard tableaux of the same shape with n boxes, the map C isobviously injective.Before we can prove the main theorem, we will need the following classicalresult about the Robinson-Schensted correspondence (see [Ful97, §4.1, Corollaryto Symmetry Theorem]): Theorem 5.9 (Symmetry Theorem for S n ) . If w ∈ S n corresponds to ( P ( w ) , Q ( w )) , then w − corresponds to ( Q ( w ) , P ( w )) under the Robinson-Schensted correspondence. Moreover, we need the following classical result by Knuth (see [Knu70, The-orem 6]): Theorem 5.10. Let x, y ∈ S n . Then x and y are Knuth equivalent if and onlyif P ( x ) = P ( y ) . In addition, we will require the following description of descent sets underthe Robinson-Schensted correspondence (see [BB05, Fact A3.4.1] and [Wil03,Proposition 2.7.1] for a proof): Definition 5.11. Let P be a standard tableau. The tableau descent set of P ,denoted by D ( P ), is the set of integers i > i + 1 lies strictly belowand weakly to the left of i in P . Lemma 5.12. For w ∈ S n we have: (i) s i ∈ L ( w ) ⇔ i ∈ D ( P ( w ))(ii) s i ∈ R ( w ) ⇔ i ∈ D ( Q ( w ))As last ingredient, we need to show that when acting on a left cell modulethe resulting structure coefficients are independent of the Q -symbols involved. Lemma 5.13. Let x, x ′ , y, y ′ ∈ S n be such that x p ∼ L y , x ′ p ∼ L y ′ , P ( x ) = P ( x ′ ) ,and P ( y ) = P ( y ′ ) . Then we have for all s ∈ S the following: p µ ys,x = p µ y ′ s,x ′ In particular, we may introduce the notation p r s ( P ( y ) , P ( x )) := p µ ys,x as thiscoefficient does not depend on the Q -symbols of x and y .Proof. First, observe that L ( x ) = L ( x ′ ) by Lemma 5.12 as the P -symbols of x and x ′ coincide. We will consider the only interesting case where s / ∈ L ( x ).By Theorem 5.10, there exists a sequence of elementary Knuth operationsrelating x and x ′ as their P -symbols coincide. Each elementary Knuth operation P -CELLS FOR SYMMETRIC GROUPS y as well. Since x and y lie in the same left cell, we have R ( x ) = R ( y )by combining Theorem 5.1 and Lemma 5.12. Therefore, the first right staroperation can be applied to y as well. By [Jen20, Theorem 4.13], applying thefirst right star operation to x and to y gives two elements that still lie in the sameleft cell. Therefore, we can repeat the argument to see that we can apply thewhole sequence of elementary Knuth transformations to y to obtain an element y ′′ . Since Knuth equivalent permutations have the same P -symbol (see Theo-rem 5.10), we have P ( y ′ ) = P ( y ) = P ( y ′′ ). Our argument also shows that x ′ and y ′′ lie in the same left cell. Therefore, we have Q ( y ′ ) = Q ( x ′ ) = Q ( y ′′ ) byTheorem 5.1. It follows that y ′ = y ′′ as both their P and Q -symbols coincide.The result now follows from repeated application of [Jen20, Corollary 4.10].Finally, we can prove our main result of this section: Theorem 5.14. The quadruple (Λ , ∗ , M, C ) gives a cell datum for H .Proof. 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