[Paper Review] Unipotent Hecke algebras of GL_n(F_q)

Nathaniel Thiem|ArXiv.org|Feb 24, 2004

Algebraic structures and combinatorial models11 references5 citations

TL;DR

This paper introduces unipotent Hecke algebras $χ_{\mu} = \mathrm{End}_G(\mathrm{Ind}_U^G(\psi_\mu))$ for $\mathrm{GL}_n(\mathbb{F}_q)$, where $U$ is the unipotent upper-triangular subgroup and $\psi_\mu$ is a linear character. It establishes a combinatorial basis indexed by matrices of monic polynomials, constructs a generalized RSK correspondence linking double coset bases to pairs of tableaux, and identifies a large commutative subalgebra isomorphic to a tensor product of smaller unipotent Hecke algebras, enabling a weight space decomposition of modules.

ABSTRACT

This paper describes a family of Hecke algebras H_μ=End_G(Ind_U^G(ψ_μ)), where U is the subgroup of unipotent upper-triangular matrices of G=GL_n(F_q) and ψ_μis a linear character of U. The main results combinatorially index a basis of H_μ, provide a large commutative subalgebra of H_μ, and after describing the combinatorics associated with the representation theory of H_μ, generalize the RSK correspondence that is typically found in the representation theory of the symmetric group.

Motivation & Objective

• To study the structure and representation theory of unipotent Hecke algebras $\mathcal{H}_\mu = \mathrm{End}_G(\mathrm{Ind}_U^G(\psi_\mu))$ for $G = \mathrm{GL}_n(\mathbb{F}_q)$, where $U$ is the unipotent radical and $\psi_\mu$ is a linear character.
• To provide a combinatorial basis for $\mathcal{H}_\mu$ indexed by $\ell \times \ell$ matrices of monic polynomials with non-vanishing constant terms and fixed row and column degree sums equal to $\mu$.
• To generalize the classical RSK correspondence by establishing a bijection between double coset representatives and pairs of tableaux indexing irreducible modules.
• To construct a large commutative subalgebra $\mathcal{L}_\mu \cong \mathcal{H}_{(\mu_1)} \otimes \cdots \otimes \mathcal{H}_{(\mu_\ell)}$ and describe a corresponding weight space decomposition of $\mathcal{H}_\mu$-modules.

Proposed method

• The standard double coset basis $\{T_v : v \in N_\mu\}$ of $\mathcal{H}_\mu$ is constructed using double cosets $U \backslash G / U$.
• A bijection is established between $N_\mu$ and the set $M_\mu$ of $\ell \times \ell$ matrices with monic polynomial entries in $\mathbb{F}_q[X]$ such that $a_{ij}(0) \neq 0$, and the sum of degrees in each row and column equals $\mu$.
• The generalized RSK correspondence is realized via a combinatorial bijection between $N_\mu$ and pairs $(P,Q)$ of column strict tableaux of the same shape $\lambda$, with weights corresponding to irreducible $\mathcal{H}_\mu$-modules.
• The commutative subalgebra $\mathcal{L}_\mu$ is constructed as $e_\mu P_\mu e_\mu$, isomorphic to the tensor product of unipotent Hecke algebras $\mathcal{H}_{(\mu_i)}$, using idempotents and group algebra projections.
• Weight space decomposition of $\mathcal{H}_\mu$-modules is defined via $\gamma$-weight spaces $V_\gamma = \{v \in V : yv = \gamma(y)v \text{ for all } y \in \mathcal{L}_\mu\}$, with dimensions computed via induced representations and Pieri's rule.
• The representation-theoretic identity $\dim(\mathcal{H}_\mu) = \sum_{\lambda \in \hat{\mathcal{H}}_\mu} \dim(\mathcal{H}_\mu^\lambda)^2$ is realized combinatorially through the RSK-type correspondence.

Experimental results

Research questions

• RQ1 How can a combinatorial basis be constructed for the unipotent Hecke algebra $\mathcal{H}_\mu$ associated to $\mathrm{GL}_n(\mathbb{F}_q)$?
• RQ2 Can the classical RSK correspondence be generalized to describe the decomposition of $\mathcal{H}_\mu$ into irreducible modules?
• RQ3 What is the structure of a large commutative subalgebra within $\mathcal{H}_\mu$, and how does it support a weight space decomposition of its modules?
• RQ4 How do the irreducible representations of $\mathcal{H}_\mu$ relate to combinatorial objects such as tableaux and partitions?
• RQ5 What is the role of the Gelfand-Graev representation and the Yokonuma algebra in the broader context of unipotent Hecke algebras?

Key findings

• A bijection is established between the double coset basis $N_\mu$ and the set $M_\mu$ of $\ell \times \ell$ matrices of monic polynomials with non-vanishing constant terms and fixed row and column degree sums equal to $\mu$.
• A generalized RSK correspondence is constructed, providing a combinatorial bijection between $N_\mu$ and pairs $(P,Q)$ of column strict tableaux of the same shape $\lambda$, where $\lambda$ indexes irreducible $\mathcal{H}_\mu$-modules.
• The dimension of $\mathcal{H}_\mu$ is realized as the sum of squares of the dimensions of its irreducible modules, with the identity $\dim(\mathcal{H}_\mu) = \sum_{\lambda \in \hat{\mathcal{H}}_\mu} \dim(\mathcal{H}_\mu^\lambda)^2$ given a combinatorial interpretation via the RSK-type correspondence.
• A large commutative subalgebra $\mathcal{L}_\mu \cong \mathcal{H}_{(\mu_1)} \otimes \cdots \otimes \mathcal{H}_{(\mu_\ell)}$ is constructed, with all irreducible $\mathcal{L}_\mu$-modules being one-dimensional.
• The dimension of the $\gamma$-weight space $V_\gamma$ in an irreducible $\mathcal{H}_\mu$-module $\mathcal{H}_\mu^\lambda$ equals the number of column strict tableaux of shape $\lambda$ and weight $\gamma$, which is $|\hat{\mathcal{H}}_\gamma^\lambda|$.
• The weight space decomposition $V \cong \bigoplus_{\gamma \in \hat{\mathcal{L}}_\mu} V_\gamma$ holds for any $\mathcal{H}_\mu$-module $V$, with dimensions computed via Frobenius reciprocity and Pieri’s rule.