[Paper Review] Graded induction for Specht modules

Jun Hu, Andrew Mathas|arXiv (Cornell University)|Aug 9, 2010

Algebraic structures and combinatorial models25 references5 citations

TL;DR

This paper proves that induced graded Specht modules for cyclotomic Hecke algebras of type G(ℓ,1,n) admit a filtration by shifts of graded Specht modules, confirming a conjecture by Brundan, Kleshchev, and Wang. Using a homogeneous cellular basis and Ryom-Hansen's ungraded filtration framework adapted to the graded setting, the authors establish a precise grading shift formula based on addable nodes and dominance order, providing a graded analog of classical induction results in representation theory.

ABSTRACT

Recently Brundan, Kleshchev and Wang introduced a $\Z$-grading on the Specht modules of the degenerate and non-degenerate cyclotomic Hecke algebras of type $G(\ell,1,n)$. In this paper we show that induced Specht modules have an explicit filtration by shifts of graded Specht modules. This proves a conjecture of Brundan, Kleshchev and Wang.

Motivation & Objective

• To prove a conjecture by Brundan, Kleshchev, and Wang on the existence of a graded Specht filtration for induced Specht modules in cyclotomic Hecke algebras of type G(ℓ,1,n).
• To extend ungraded induction results to the graded setting by constructing a filtration of induced modules using graded Specht modules with explicit degree shifts.
• To establish a precise relationship between the combinatorics of addable nodes and the grading shifts in the induced module filtration, using dominance order and node positions.

Proposed method

• Adapt Ryom-Hansen's ungraded filtration construction to the graded setting using a homogeneous cellular basis for the cyclotomic quiver Hecke algebra H^Λ_n.
• Define the graded induction functor i-Ind via tensor product with the inclusion H^Λ_n → H^Λ_{n+1}, decomposed over the index set I = Z/eZ.
• Construct a basis B_{μ,i} for the image of the idempotent z_μ↑i in H^Λ_{n+1}, indexed by standard tableaux on multipartitions obtained by adding addable i-nodes.
• Use the transition matrix results between standard and homogeneous bases (Theorem 3.7) to derive non-vanishing conditions for products of basis elements, ensuring filtration components are non-zero.
• Define a filtration 0 = I_0 ⊂ I_1 ⊂ ... ⊂ I_z = i-Ind S_μ by ordering addable i-nodes downward, with I_j/I_{j-1} isomorphic to S_{α_j}⟨d_{A_j}(μ)⟩.
• Verify that the grading shifts match the predicted formula: def β + codeg t_μ + d_{A_j}(μ), using known degree identities from [8, Lemma 3.12].

Experimental results

Research questions

• RQ1 Does the induced graded Specht module i-Ind S_μ admit a filtration by shifts of graded Specht modules, as conjectured by Brundan, Kleshchev, and Wang?
• RQ2 What is the precise formula for the grading shift d_{A_j}(μ) in terms of the addable node A_j and the multipartition μ?
• RQ3 How can Ryom-Hansen's ungraded filtration construction be adapted to the graded setting using cellular basis techniques and quiver Hecke algebra structure?
• RQ4 What is the role of the dominance order on multipartitions in determining the filtration order of the induced module?
• RQ5 Can the graded dual Specht module i-Ind S_μ′ be similarly filtered, and what is the corresponding shift formula?

Key findings

• The induced graded Specht module i-Ind S_μ admits a graded Specht filtration: 0 = I_0 ⊂ I_1 ⊂ ... ⊂ I_z = i-Ind S_μ with I_j/I_{j-1} ≅ S_{α_j}⟨d_{A_j}(μ)⟩.
• The filtration order corresponds to the dominance order on multipartitions obtained by adding addable i-nodes in 'downwards' order, with α_1 being the most dominant and α_z the least dominant.
• The grading shift d_{A_j}(μ) is determined by the relative position of the addable node A_j in the multipartition μ, with d_{A_j}(μ) = deg t_{α_j}^μ - deg t_μ.
• The total degree shift in the filtration is given by def β + codeg t_μ + d_{A_j}(μ), where def β and codeg t_μ are known invariants from [8, Lemma 3.12].
• The result extends to the dual Specht modules: i-Ind S_μ′ has a filtration with quotients S_{α_k}⟨d_{A_k}(μ)⟩, where the addable nodes are ordered in reverse dominance order.
• The proof relies on a homogeneous cellular basis and non-vanishing conditions for products of basis elements, ensuring the filtration components are well-defined and non-zero.