[Paper Review] Harish-Chandra series in finite unitary groups and crystal graphs

Thomas Gerber, Gerhard Hiß+1 more|arXiv (Cornell University)|Aug 6, 2014

Algebraic structures and combinatorial models21 references4 citations

TL;DR

This paper proposes a conjectural connection between the Harish-Chandra series of unipotent modules in finite unitary groups and the crystal graph structure of Fock spaces for quantum groups. By generalizing Harish-Chandra theory to include pure Levi subgroups and introducing weakly cuspidal modules, the authors link modular representation theory with combinatorics via crystal isomorphisms, showing that distinct $ε$-cores in partitions correspond to distinct $ε$-blocks under the conjecture, with supporting evidence from Iwahori-Hecke algebras and $β$-set manipulations.

ABSTRACT

The distribution of the unipotent modules (in non-defining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph of an integrable module for a certain quantum group. Evidence for our conjectures is presented, as well as proofs for some of their consequences for the crystal graphs involved. In the course of our work we also generalize Harish-Chandra theory for some of the finite classical groups, and we introduce their Harish-Chandra branching graphs.

Motivation & Objective

• To generalize Harish-Chandra theory for finite unitary, symplectic, and odd-dimensional orthogonal groups by restricting to pure Levi subgroups.
• To define weakly cuspidal modules and establish a refined Harish-Chandra series decomposition for unipotent modules in non-defining characteristic.
• To formulate a conjectural correspondence between the $\ell$-modular Harish-Chandra series of unipotent modules and the crystal graph of an integrable module for a quantum group.
• To provide evidence for this conjecture using endomorphism rings of induced modules and combinatorial tools like $\u03b2$-sets and $e$-cores.
• To establish that distinct $e$-cores in partitions imply distinct $\ell$-blocks for simple submodules in branching rules, under the conjecture.

Proposed method

• Generalize Harish-Chandra theory by restricting induction to pure Levi subgroups, defined by connected subsets of the Dynkin diagram containing the first node adjacent to the double edge.
• Define weakly cuspidal modules as cuspidal modules for these pure Levi subgroups and show that the usual Harish-Chandra series are unions of weak Harish-Chandra series.
• Prove that the endomorphism ring of a Harish-Chandra induced weakly cuspidal module is isomorphic to an Iwahori-Hecke algebra of type $B$, with parameters related by reduction modulo $\ell$ under certain conditions.
• Construct the Harish-Chandra branching graph for unipotent modules, recording socle composition factors of induced modules, analogous to Kleshchev’s branching rules for symmetric groups.
• Use $\u03b2$-sets and their transformations under Kashiwara operators to model crystal isomorphisms and relate the structure of the crystal graph to the $e$-core of partitions.
• Apply the canonical crystal isomorphism and charge shifting to relate different realizations of the same crystal graph, particularly in the context of $\mathfrak{B}((-,-),\mathbf{d})$.

Experimental results

Research questions

• RQ1 How can Harish-Chandra theory be generalized to finite classical groups like unitary, symplectic, and odd orthogonal groups using pure Levi subgroups?
• RQ2 What is the precise relationship between the $\ell$-modular Harish-Chandra series of unipotent modules and the crystal graph of a Fock space for a quantum group?
• RQ3 Under what conditions do two unipotent modules in the same $\ell$-modular weak Harish-Chandra series have the same $2$-core?
• RQ4 How do the $e$-cores of partitions labeling unipotent modules relate to their $\ell$-block membership in the branching of induced modules?
• RQ5 Can the combinatorics of $\u03b2$-sets and Kashiwara operators be used to prove that distinct $e$-cores imply distinct $\ell$-blocks for simple submodules in branching rules?

Key findings

• The endomorphism ring of a Harish-Chandra induced weakly cuspidal module is isomorphic to an Iwahori-Hecke algebra of type $B$, with parameters related by reduction modulo $\ell$ when the block contains an ordinary cuspidal module.
• Conjecture 5.4 predicts that if two unipotent modules in characteristic $\ell$ lie in the same weak Harish-Chandra series, then their corresponding partitions have the same $2$-core, refining the ordinary Harish-Chandra series.
• The Harish-Chandra branching graph for unipotent modules records the socle composition factors of induced modules, generalizing Kleshchev’s branching rules for symmetric groups.
• For $\ell$ large enough and under Conjecture 5.7, any two non-isomorphic simple submodules of $R_{\mathrm{GU}_n(q)}^{\mathrm{GU}_{n+2}(q)}(X)$ lie in distinct $\ell$-blocks, as their corresponding partitions have distinct $e$-cores.
• Crystal isomorphisms between $B(\mu,\mathbf{c})$ and $B((-,-),\mathbf{d})$ are established via charge shifting and $\u03b2$-set transformations, with the difference in charge components matching $s + (1-e)/2$ or $s + (1+e)/2$.
• The proof of the proposition on distinct $e$-cores relies on the fact that applying different Kashiwara operators $\widetilde{f}_{j_1}$ and $\widetilde{f}_{j_2}$ to the same module results in $\beta$-sets that cannot yield the same symbol after $e$-core reduction when $j_1 \not\equiv j_2 \pmod{e}$.