Equivariant incidence algebras and equivariant Kazhdan-Lusztig-Stanley theory
aa r X i v : . [ m a t h . C O ] S e p Equivariant incidence algebras and equivariant Kazhdan–Lusztig–Stanley theory
Nicholas Proudfoot
Department of Mathematics, University of Oregon, Eugene, OR [email protected]
Abstract.
We establish a formalism for working with incidence algebras of posets with symme-tries, and we develop equivariant Kazhdan–Lusztig–Stanley theory within this formalism. Thisgives a new way of thinking about the equivariant Kazhdan–Lusztig polynomial and equivariant Z -polynomial of a matroid. The incidence algebra of a locally finite poset was first introduced by Rota, and has proved to bea natural formalism for studying such notions as M¨obius inversion [Rot64], generating functions[DRS72], and Kazhdan–Lusztig–Stanley polynomials [Sta92, Section 6].A special class of Kazhdan–Lusztig–Stanley polynomials that have received a lot of attentionrecently is that of Kazhdan–Lusztig polynomials of matroids, where the relevant poset is the latticeof flats [EPW16, Pro18]. If a finite group W acts on a matroid M (and therefore on the latticeof flats), one can define the W -equivariant Kazhdan–Lusztig polynomial of M [GPY17]. This is apolynomial whose coefficients are virtual representations of W , and has the property that takingdimensions recovers the ordinary Kazhdan–Lusztig polynomial of M . In the case of the uniformmatroid of rank d on n elements, it is actually much easier to describe the S n -equivariant Kazhdan–Lusztig polynomial, which admits a nice description in terms of partitions of n , than it is to describethe non-equivariant Kazhdan–Lusztig polynomial [GPY17, Theorem 3.1].While the definition of Kazhdan–Lusztig–Stanley polynomials is greatly clarified by the lan-guage of incidence algebras, the definition of the equivariant Kazhdan–Lusztig polynomial of amatroid is completely ad hoc and not nearly as elegant. The purpose of this note is to define theequivariant incidence algebra of a poset with a finite group of symmetries, and to show that thebasic constructions of Kazhdan–Lusztig–Stanley theory make sense in this more general setting. Inthe case of a matroid, we show that this approach recovers the same equivariant Kazhdan–Lusztigpolynomials that were defined in [GPY17]. Acknowledgments:
We thank Tom Braden for his feedback on a preliminary draft of this work.
Fix once and for all a field k . Let P be a locally finite poset equipped with the action of a finitegroup W . We consider the category C W ( P ) whose objects consist of1 a k -vector space V • a direct product decomposition V = Q x ≤ y ∈ P V xy , with each V xy finite dimensional • an action of W on V compatible with the decomposition.More concretely, for any σ ∈ W and any x ≤ y ∈ P , we have a linear map ϕ σxy : V xy → V σ ( x ) σ ( y ) ,and we require that ϕ exy = id V xy and that ϕ σ ′ σ ( x ) σ ( y ) ◦ ϕ σxy = ϕ σ ′ σxy . Morphisms in C W ( P ) are definedto be linear maps that are compatible with both the decomposition and the action. This categoryadmits a monoidal structure, with tensor product given by( U ⊗ V ) xz := M x ≤ y ≤ z U xy ⊗ V yz . Let I W ( P ) be the Grothendieck ring of C W ( P ); we call I W ( P ) the equivariant incidence algebra of P with respect to the action of W . Example 2.1. If W is the trivial group, then I W ( P ) is isomorphic to the usual incidence algebraof P with coefficients in Z . That is, it is isomorphic as an abelian group to a direct product ofcopies of Z , one for each interval in P , and multiplication is given by convolution. Remark 2.2. If W acts on P and ψ : W ′ → W is a group homomorphism, then ψ induces afunctor F ψ : C W ( P ) → C W ′ ( P ) and a ring homomorphism R ψ : I W ( P ) → I W ′ ( P ).We now give a second, more down to earth description of I W ( P ). Let VRep( W ) denote thering of finite dimensional virtual representations of W over the field k . A group homomorphism ψ : W ′ → W induces a ring homomorphism Λ ψ : VRep( W ) → VRep( W ′ ). For any x ∈ P , let W x ⊂ W be the stabilizer of x . We also define W xy := W x ∩ W y and W xyz := W x ∩ W y ∩ W z . Notethat, for any x, y ∈ P and σ ∈ W , conjugation by σ gives a group isomorphism ψ σxy : W xy → W σ ( x ) σ ( y ) , which induces a ring isomorphismΛ ψ σxy : VRep( W σ ( x ) σ ( y ) ) → VRep( W xy ) . An element f ∈ I W ( P ) is uniquely determined by a collection { f xy | x ≤ y ∈ P } , where f xy ∈ VRep( W xy ) and for any σ ∈ W and x ≤ y ∈ P , f xy = Λ ψ σxy (cid:0) f σ ( x ) σ ( y ) (cid:1) . The unit δ ∈ I W ( P ) is characterized by the property that δ xx is the 1-dimensional trivial representation of W x for all x ∈ P and δ xy = 0 for all x < y ∈ P . The following proposition describes the productstructure on I W ( P ) in this representation. 2 roposition 2.3. For any f, g ∈ I W ( P ) . ( f g ) xz := X x ≤ y ≤ z | W xyz || W xz | Ind W xz W xyz (cid:16)(cid:16) Res W xy W xyz f xy (cid:17) ⊗ (cid:16) Res W yz W xyz g yz (cid:17)(cid:17) . Remark 2.4.
It may be surprising to see the fraction | W xyz || W xz | in the statement of Proposition 2.3,since VRep( W xy ) is not a vector space over the rational numbers. We could in fact replace the sumover [ x, z ] with a sum over one representative of each W xz -orbit in [ x, z ] and then eliminate the factorof | W xyz || W xz | . Including the fraction in the equation allows us to avoid choosing such representatives. Remark 2.5.
Proposition 2.3 could be taken as the definition of I W ( P ). It is not so easy toprove associativity directly from this definition, though it can be done with the help of Mackey’srestriction formula (see for example [Bum13, Corollary 32.2]). Remark 2.6.
Suppose that ψ : W ′ → W is a group homomorphism, and for any x, y ∈ P ,consider the induced group homomorphism ψ xy : W ′ xy → W xy . For any f ∈ I W ( P ), we have, R ψ ( f ) xy = Λ ψ xy ( f xy ). In particular, if W ′ is the trivial group, then R ψ ( f ) xy is equal to thedimension of the virtual representation f xy ∈ VRep( W xy ).Before proving Proposition 2.3, we state the following standard lemma in representation theory. Lemma 2.7.
Suppose that E = L s ∈ S E s is a vector space that decomposes as a direct sum ofpieces indexed by a finite set S . Suppose that G acts linearly on E and acts by permutations on S such that, for all s ∈ S and γ ∈ G , γ · E s = E γ · s . For each x ∈ S , let G x ⊂ G denote the stabilizerof s . Then there exists an isomorphism E ∼ = M s ∈ S | G s || G | Ind GG s (cid:0) E s (cid:1) of representations of G . Proof of Proposition 2.3.
By linearity, it is sufficient to prove the proposition in the case where wehave objects U and V of C W ( P ) with f = [ U ] and g = [ V ]. This means that, for all x ≤ y ≤ z ∈ P , f xy = [ U xy ] ∈ VRep( W xy ), g yz = [ V yz ] ∈ VRep( W yz ), and( f g ) xz = (cid:2) ( U ⊗ V ) xz (cid:3) = M x ≤ y ≤ z U xy ⊗ V yz ∈ VRep( W xz ) . The proposition then follows from Lemma 2.7 by taking E = ( U ⊗ V ) xz , S = [ x, z ], and G = W xz .Let R be a commutative ring. Given an element f ∈ I W ( P ) ⊗ R and a pair of elements x ≤ y ∈ P ,we will write f xy to denote the corresponding element of VRep( W xy ) ⊗ R . As in Remark 2.4, we may eliminate the fraction at the cost of choosing one representative of each W -orbit in S . roposition 2.8. An element f ∈ I W ( P ) ⊗ R is (left or right) invertible if and only if f xx ∈ VRep( W x ) ⊗ R is invertible for all x ∈ P . In this case, the left and right inverses are unique andthey coincide.Proof. By Proposition 2.3, an element g is a right inverse to f if and only if g xx = f − xx for all x ∈ P and X x ≤ y ≤ z | W xyz || W xz | Ind W xz W xyz (cid:16)(cid:16) Res W xy W xyz f xy (cid:17) ⊗ (cid:16) Res W yz W xyz g yz (cid:17)(cid:17) = 0for all x < z ∈ P . The second condition can be rewritten as (cid:16)
Res W x W xz f xx (cid:17) ⊗ g xz = − X x Let ζ ∈ I W ( P ) be the element defined by letting ζ xy ( t ) be the trivial representationof W xy in degree zero for all x ≤ y , and let χ := ζ − ¯ ζ . The function χ is called the equivariantcharacteristic function of P with respect to the action of W . We have χ − = ¯ ζ − ζ = ¯ χ , so χ is a P -kernel. Since ¯ ζ = ζχ , ζ is equal to the left KLS-function associated with χ . However, theright KLS-function f associated with χ is much more interesting! See Propositions 4.1 and 4.3 fora special case of this construction.We next introduce the equivariant analogue of the material in [Pro18, Section 2.3]. If κ is a P -kernel with right and left KLS-functions f and g , we define Z := gκf ∈ I W ( P ), which we callthe equivariant Z -function associated with κ . For any x ≤ y , we will refer to the graded virtualrepresentation Z xy ( t ) as an equivariant Z -polynomial . Proposition 3.3. We have ¯ Z = Z .Proof. Since ¯ g = gκ , we have Z = gκf = ¯ gf . Since ¯ f = κf , we have Z = gκf = g ¯ f . Thus¯ Z = ¯ gf = ¯¯ g ¯ f = g ¯ f = Z . Remark 3.4. Suppose that κ ∈ I W ( P ) is a P -kernel and f, g, Z ∈ I W ( P ) are the associatedequivariant KLS-functions and equivariant Z -function. It is immediate from the definitions that,if ψ : W ′ → W is a group homomorphism, then R ψ ( f ) , R ψ ( g ) , R ψ ( Z ) ∈ I W ′ ( P ) are the equiv-ariant KLS-functions and equivariant Z -function associated with the P -kernel R ψ ( κ ) ∈ I W ′ ( P ).In particular, if we take W ′ to be the trivial group, then Remark 2.6 tells us that the ordinaryKLS-polynomials and Z -polynomials are recovered from the equivariant KLS-polynomials and Z -polynomials by sending virtual representations to their dimensions. Let M be a matroid, let L be the lattice of flats of M equipped with the usual weak rank function,and let W be a finite group acting on L . Let OS WM ( t ) be the Orlik–Solomon algebra of M , regarded6s a graded representation of W . Following [GPY17, Section 2], we define H WM ( t ) := t rk M OS WM ( − t − ) ∈ VRep( W ) ⊗ Z [ t ] . If W is trivial, then H WM ( t ) ∈ Z [ t ] is equal to the characteristic polynomial of M . For any F ≤ G ∈ L ,let M F G be the minor of M with lattice of flats [ F, G ] obtained by deleting the complement of G and contracting F ; this matroid inherits an action of the stabilizer group W F G ⊂ W . Define H ∈ I W ( L ) by putting H F G ( t ) = H W F G M F G ( t ) for all F ≤ G . Proposition 4.1. The function H is equal to the equivariant characteristic function of L .Proof. It is proved in [GPY17, Lemma 2.5] that ζH = ¯ ζ . Multiplying on the left by ζ − , we have H = ζ − ¯ ζ , which is the definition of the equivariant characteristic function of L . Remark 4.2. The proof of [GPY17, Lemma 2.5] is surprisingly difficult. Consequently, Proposi-tion 4.1 is a deep fact about Orlik–Solomon algebras, not just a formal consequence of the defini-tions.The equivariant Kazhdan–Lusztig polynomial P WM ( t ) ∈ VRep( W ) ⊗ Z [ t ] was introducedin [GPY17, Section 2.2]. Define P ∈ I W / ( L ) by putting P F G ( t ) = P W F G M F G ( t ) for all F ≤ G . Thedefining recursion for P WM ( t ) in [GPY17, Theorem 2.8] translates to the formula ¯ P = HP , whichimmediately implies the following proposition. Proposition 4.3. The function P is the right equivariant KLS-function associated with H . The equivariant Z -polynomial Z WM ( t ) ∈ VRep( W ) ⊗ Z [ t ] was introduced in [PXY18, Section6]. Define Z ∈ I W ( L ) by putting Z F G ( t ) = Z W F G M F G ( t ) for all F ≤ G . The defining recursion for Z WM ( t ) in [PXY18, Section 6] translates to the formula Z = ¯ ζP . Proposition 4.4. The function Z is the Z -function associated with H .Proof. Example 3.2 tells us that the right KLS-function associated with H is ζ and Proposition4.3 tells us that the left KLS-function associated with H is P , thus the Z -function is equal ζHP =¯ ζP = Z. The following corollary was asserted without proof in [PXY18, Section 6], and follows immedi-ately from Propositions 3.3 and 4.4. Corollary 4.5. The polynomial Z WM ( t ) is palindromic. That is, t rk M Z WM ( t − ) = Z WM ( t ) . When W is the trivial group, Gao and Xie define polynomials Q M ( t ) and ˆ Q M ( t ) = ( − rk M Q M ( t )with the property that (cid:0) P − (cid:1) F G ( t ) = ˆ Q M F G ( t ) [GX20]. If ˆ0 and ˆ1 are the minimal and maximalflats of M , this is equivalent to the statement that Q M ( t ) = ( − rk M (cid:0) P − (cid:1) ˆ0ˆ1 ( t ). The polynomial Q M ( t ) is called the inverse Kazhdan–Lusztig polynomial of M . Using the machinery of The difficult part appears in the proof of Lemma 2.4, which is then used to prove Lemma 2.5. The reason for bestowing this name on Q M ( t ) rather than ˆ Q M ( t ) is that Q M ( t ) has non-negative coefficients. equivariantinverse Kazhdan–Lusztig polynomial Q WM ( t ) := ( − rk M (cid:0) P − (cid:1) ˆ0ˆ1 ( t ) . If we then define ˆ Q ∈ I W / ( L ) by putting ˆ Q F G ( t ) = ( − r F G Q W F G M F G ( t ) for all F ≤ G , we immediatelyobtain the following proposition. Proposition 4.6. The functions P and ˆ Q are mutual inverses in I W ( L ) . References [Bre99] Francesco Brenti, Twisted incidence algebras and Kazhdan-Lusztig-Stanley functions ,Adv. Math. (1999), no. 1, 44–74.[Bum13] Daniel Bump, Lie groups , second ed., Graduate Texts in Mathematics, vol. 225, Springer,New York, 2013.[DRS72] Peter Doubilet, Gian-Carlo Rota, and Richard Stanley, On the foundations of combinato-rial theory. VI. The idea of generating function , Proceedings of the Sixth Berkeley Sym-posium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif.,1970/1971), Vol. II: Probability theory, 1972, pp. 267–318.[EPW16] Ben Elias, Nicholas Proudfoot, and Max Wakefield, The Kazhdan-Lusztig polynomial ofa matroid , Adv. Math. (2016), 36–70.[GPY17] Katie Gedeon, Nicholas Proudfoot, and Benjamin Young, The equivariant Kazhdan–Lusztig polynomial of a matroid , J. Combin. Theory Ser. A (2017), 267–294.[GX20] Alice L. L. Gao and Matthew H.Y. Xie, The inverse Kazhdan-Lusztig polynomial of amatroid , 2020, arXiv:2007.15349 .[Pro18] Nicholas Proudfoot, The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials ,EMS Surv. Math. Sci. (2018), no. 1, 99–127.[PXY18] Nicholas Proudfoot, Yuan Xu, and Ben Young, The Z -polynomial of a matroid , Electron.J. Combin. (2018), no. 1, Paper 1.26, 21.[Rot64] Gian-Carlo Rota, On the foundations of combinatorial theory. I. Theory of M¨obius func-tions , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete (1964), 340–368 (1964).[Sta92] Richard P. Stanley, Subdivisions and local h -vectors , J. Amer. Math. Soc.5