Equivariant incidence algebras and equivariant Kazhdan-Lusztig-Stanley theory

We establish a formalism for working with incidence algebras of posets with symmetries, and we develop equivariant Kazhdan-Lusztig-Stanley theory within this formalism. This gives a new way of thinking about the equivariant Kazhdan-Lusztig polynomial and equivariant Z-polynomial of a matroid.


Introduction
The incidence algebra of a locally finite poset was first introduced by Rota, and has proved to be a natural formalism for studying such notions as Möbius 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 attention recently is that of Kazhdan-Lusztig polynomials of matroids, where the relevant poset is the lattice of flats [EPW16,Pro18]. If a finite group W acts on a matroid M (and therefore on the lattice of flats), one can define the W -equivariant Kazhdan-Lusztig polynomial of M [GPY17]. This is a polynomial whose coefficients are virtual representations of W , and has the property that taking dimensions recovers the ordinary Kazhdan-Lusztig polynomial of M . In the case of the uniform matroid 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 describe the non-equivariant Kazhdan-Lusztig polynomial [GPY17, Theorem 3.1].
While the definition of Kazhdan-Lusztig-Stanley polynomials is greatly clarified by the language of incidence algebras, the definition of the equivariant Kazhdan-Lusztig polynomial of a matroid is completely ad hoc and not nearly as elegant. The purpose of this note is to define the equivariant incidence algebra of a poset with a finite group of symmetries, and to show that the basic constructions of Kazhdan-Lusztig-Stanley theory make sense in this more general setting. In the case of a matroid, we show that this approach recovers the same equivariant Kazhdan-Lusztig polynomials that were defined in [GPY17].
• a k-vector space V • a direct product decomposition V = 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 σ xy : V xy → V σ(x)σ(y) , and we require that ϕ e xy = id Vxy and that ϕ σ ′ σ(x)σ(y) • ϕ σ xy = ϕ σ ′ σ xy . Morphisms in C W (P ) are defined to be linear maps that are compatible with both the decomposition and the action. This category admits a monoidal structure, with tensor product given by 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 algebra of P with coefficients in Z. That is, it is isomorphic as an abelian group to a direct product of copies 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 a functor 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 the ring 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 . Note that, for any x, y ∈ P and σ ∈ W , conjugation by σ gives a group isomorphism which induces a ring isomorphism .
where f xy ∈ VRep(W xy ) and for any σ ∈ W and x ≤ y ∈ P , f xy = Λ ψ σ xy f σ(x)σ(y) . 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 product structure on I W (P ) in this representation. Proposition 2.3. For any f, g ∈ I W (P ).
Remark 2.4. It may be surprising to see the fraction |Wxyz| |Wxz| 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 sum over [x, z] with a sum over one representative of each W xz -orbit in [x, z] and then eliminate the factor of |Wxyz| |Wxz| . 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 to prove associativity directly from this definition, though it can be done with the help of Mackey's restriction 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 ψxy (f xy ). In particular, if W ′ is the trivial group, then R ψ (f ) xy is equal to the dimension 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 = s∈S E s is a vector space that decomposes as a direct sum of pieces 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 stabilizer of s. Then there exists an isomorphism Proof of Proposition 2.3. By linearity, it is sufficient to prove the proposition in the case where we have objects U and V of C W (P ) with f = [U ] and g = [V ]. This means that, for all x ≤ y ≤ z ∈ P , 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. for all x < z ∈ P . 2 The second condition can be rewritten as and this equation has a unique solution for g. Thus f has a right inverse if and only if f xx ∈ VRep(W x ) ⊗ R is invertible for all x ∈ P . The argument for left inverses is identical, so it remains only to show that left and right inverses coincide.
Let g be right inverse to f . Then g is also left inverse to some function, which we will denote h. We then have so g is left inverse to f , as well.

Equivariant Kazhdan-Lusztig-Stanley theory
In this section we take R to be the ring Z[t] and for each f ∈ I W (P ) ⊗ Z[t] and x ≤ y ∈ P , we write f xy (t) for the corresponding component of f . One can regard f xy (t) as a polynomial whose coefficients are virtual representations of W xy , or equivalently as a graded virtual representation of W xy . We assume that P is equipped with a W -invariant weak rank function in the sense of [Bre99, Section 2]. This is a collection of natural numbers {r xy ∈ N | x ≤ y ∈ P } with the following properties: x ≤ y and σ ∈ W . Note that I W (P ) is a subalgebra of I W (P ), and we define an involution f →f of I W (P ) by puttingf xy (t) := t rxy f xy (t −1 ). An element κ ∈ I W (P ) is called a P -kernel if κ xx (t) = δ xx (t) for all x ∈ P andκ = κ −1 .
Proof. We follow the proof in [Pro18, Theorem 2.2]. We will prove existence and uniqueness of f ; the proof for g is identical. Fix elements x < w ∈ P . Suppose that f yw (t) has been defined for all x < y ≤ w and that the equationf = κf holds where defined. Let The equationf = κf for the interval [x, w] translates tō It is clear that there is at most one polynomial f xw (t) of degree strictly less than r xw /2 satisfying this equation. The existence of such a polynomial is equivalent to the statement To prove this, we observe that This is formally equal to the expression for (κ(κf )) xw − (κf ) xw , which by associativity is equal to the expression for Thus we have Thus there is a unique choice of polynomial f xw (t) consistent with the equationf = κf on the interval [x, w].
We will refer to the element f ∈ I W 1 /2 (P ) from Theorem 3.1 is the right equivariant KLSfunction associated with κ, and to g as the left equivariant KLS-function associated with κ.
For any x ≤ y, we will refer to the graded virtual representations f xy (t) and g xy (t) as (right or left) equivariant KLS-polynomials. When W is the trivial group, these definitions specialize to the ones in [Pro18, Section 2].
Example 3.2. Let ζ ∈ I W (P ) be the element defined by letting ζ xy (t) be the trivial representation of W xy in degree zero for all x ≤ y, and let χ := ζ −1ζ . The function χ is called the equivariant characteristic function of P with respect to the action of W . We have χ −1 =ζ −1 ζ =χ, so χ is a P -kernel. Sinceζ = ζχ, ζ is equal to the left KLS-function associated with χ. However, the right KLS-function f associated with χ is much more interesting! See Propositions 4.1 and 4.3 for a 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 call the equivariant Z-function associated with κ. For any x ≤ y, we will refer to the graded virtual representation Z xy (t) as an equivariant Z-polynomial.
Remark 3.4. Suppose that κ ∈ I W (P ) is a P -kernel and f, g, Z ∈ I W (P ) are the associated equivariant 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 equivariant 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 ordinary KLS-polynomials and Z-polynomials are recovered from the equivariant KLS-polynomials and Zpolynomials by sending virtual representations to their dimensions.

Matroids
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. If we then defineQ ∈ I W 1 /2 (L) by puttingQ F G (t) = (−1) r F G Q W F G M F G (t) for all F ≤ G, we immediately obtain the following proposition.
Proposition 4.6. The functions P andQ are mutual inverses in I W (L).