[Paper Review] FI_W-modules and constraints on classical Weyl group characters

J. Wilson|arXiv (Cornell University)|Mar 30, 2015

Advanced Combinatorial Mathematics44 references7 citations

TL;DR

This paper introduces FI_W-modules to study sequences of representations of classical Weyl groups (symmetric, hyperoctahedral, and even-signed groups), proving that finitely generated FI_W-modules over characteristic zero have character polynomials independent of n for large n. This implies polynomial growth in dimension and uniform description of characters via signed-cycle-counting functions, extending prior results on symmetric groups to types B/C and D.

ABSTRACT

In this paper we study the characters of sequences of representations of any of the three families of classical Weyl groups W_n: the symmetric groups, the signed permutation groups (hyperoctahedral groups), or the even-signed permutation groups. Our results extend work of Church, Ellenberg, Farb, and Nagpal on the symmetric groups. We use the concept of an FI_W-module, an algebraic object that encodes the data of a sequence of W_n-representations with maps between them, defined in the author's recent work ArXiv:1309.3817. We show that if a sequence {V_n} of W_n-representations has the structure of a finitely generated FI_W-module, then there are substantial constraints on the growth of the sequence and the structure of the characters: for n large, the dimension of V_n is equal to a polynomial in n, and the characters of V_n are given by a character polynomial in signed-cycle-counting class functions, independent of n. We determine bounds the degrees of these polynomials. We continue to develop the theory of FI_W-modules, and we apply this theory to obtain new results about a number of sequences associated to the classical Weyl groups: the cohomology of complements of classical Coxeter hyperplane arrangements, and the cohomology of the pure string motion groups (the groups of symmetric automorphisms of the free group).

Motivation & Objective

• To extend the theory of FI-modules from symmetric groups to classical Weyl groups of types B/C and D.
• To establish that finitely generated FI_W-modules over characteristic zero have character polynomials independent of n for large n.
• To prove that the dimensions of such representations grow polynomially in n, even over arbitrary fields.
• To develop the theory of FI_W#-modules, a subclass with stronger symmetries, to deduce tighter structural constraints.
• To apply the framework to cohomology sequences of hyperplane complements and pure string motion groups.

Proposed method

• Define FI_W-modules as algebraic structures encoding sequences of W_n-representations with compatible maps.
• Use the theory of induced representations M_W(U) to classify FI_W-modules and analyze their character polynomials.
• Prove that characters of V_n for large n are given by character polynomials in signed-cycle-counting class functions, with degree bounded by the finite generation degree.
• Establish polynomial dimension growth over arbitrary fields via a generalization of the CEFN14 result to type B/C and D.
• Introduce FI_W#-modules as a refinement with additional symmetries, enabling stronger constraints on representation decompositions.
• Apply the framework to compute character polynomials for cohomology of hyperplane complements M_W(n) and pure string motion groups PΣ_n.

Experimental results

Research questions

• RQ1 What constraints does finite generation of an FI_W-module impose on the characters of its representations V_n for large n?
• RQ2 Can the dimension of V_n in a finitely generated FI_W-module grow polynomially even over arbitrary fields?
• RQ3 How do character polynomials in types B/C and D generalize the known results from type A (symmetric groups)?
• RQ4 What is the structure of the cohomology of the complement of classical Coxeter hyperplane arrangements as an FI_W-module?
• RQ5 How do FI_W#-modules refine the structure of FI_W-modules and lead to stronger constraints on representation decompositions?

Key findings

• For any finitely generated FI_W-module over characteristic zero, the character χ_V_n(σ) is given by a character polynomial F_V of degree at most d, independent of n, for all σ ∈ W_n and all n sufficiently large.
• The dimension dim(V_n) is equal to a polynomial in n for all n sufficiently large, with the degree bounded by the finite generation degree d.
• This polynomial growth in dimension holds over arbitrary fields, not just characteristic zero, as shown by Theorem 4.20.
• The character polynomial for H^2(M_BC(•), ℂ) is explicitly computed as 3(X_1 choose 2) + 3(Y_1 choose 2) - X_1Y_1 + 3X_2 - Y_2 + 14(X_1 choose 3) + 2(X_1 choose 2)Y_1 + ...
• The decomposition of H^m(M_W(•), ℂ) into induced representations M_W(U) is computable via the character polynomial, as posed in Problem 5.11.
• FI_W#-modules allow for stronger structural constraints on the representation decompositions of V_n, enabling more refined analysis than standard FI_W-modules.