[Paper Review] Specht modules and Kazhdan--Lusztig cells in type $B_n$

Meinolf Geck, Lăcrimioara Iancu+1 more|ArXiv.org|Apr 16, 2007

Algebraic structures and combinatorial models4 references3 citations

TL;DR

This paper establishes a canonical isomorphism between Specht modules and Kazhdan–Lusztig left cell modules for the Hecke algebra of type $B_n$ in the asymptotic case, where the parameters are ordered via the dominance order. The isomorphism is shown to be triangular with 1s on the diagonal when bases are suitably ordered, and the result fails for other monomial orders, as demonstrated by counterexamples.

ABSTRACT

Dipper, James and Murphy generalized the classical Specht module theory to Hecke algebras of type $B_n$. On the other hand, for any choice of a monomial order on the parameters in type $B_n$, we obtain corresponding Kazhdan--Lusztig cell modules. In this paper, we show that the Specht modules are naturally equivalent to the Kazhdan--Lusztig cell modules {\em if} we choose the dominance order on the parameters, as in the ``asymptotic case'' studied by Bonnafé and the second named author. We also give examples which show that such an equivalence does not hold for other choices of monomial orders.

Motivation & Objective

• To establish a natural isomorphism between Specht modules and Kazhdan–Lusztig cell modules in type $B_n$.
• To identify the specific parameter ordering—namely the dominance (asymptotic) case—under which this isomorphism holds.
• To demonstrate that such an isomorphism does not exist for other monomial orders on the parameters.
• To provide explicit combinatorial descriptions of distinguished left cells in the asymptotic case using reduced expressions.
• To compare the cellular structure of Kazhdan–Lusztig bases with the Specht module construction in the specialized algebra setting.

Proposed method

• Utilizes the combinatorial description of left cells in the asymptotic case from Bonnafé and Iancu (2006), which allows explicit identification of distinguished left cells for each bipartition.
• Constructs a canonical isomorphism between the Specht module $ ilde{S}^{ u}$ and the Kazhdan–Lusztig left cell module $W_{k}( u)$ in the asymptotic case.
• Shows that the matrix of the isomorphism is triangular with 1s on the diagonal when both modules are equipped with standard bases ordered appropriately.
• Employs the Kazhdan–Lusztig basis ${ C_w }$ of the Hecke algebra $\mathcal{H}$, defined via the Bruhat–Chevalley order and structure constants $P_{y,w}^*$ in $A_{<0}$.
• Uses the cellular algebra structure of $\mathcal{H}_{n,k}$ under the asymptotic condition $b > (n-1)a > 0$, ensuring compatibility between Specht and cell modules.
• Provides counterexamples in Section 4 to show that the isomorphism fails for non-asymptotic monomial orders, such as the weighted lexicographic order.

Experimental results

Research questions

• RQ1 Under what parameter ordering in type $B_n$ do Specht modules and Kazhdan–Lusztig cell modules become isomorphic?
• RQ2 Is the isomorphism between Specht and Kazhdan–Lusztig cell modules canonical and triangular with 1s on the diagonal in the asymptotic case?
• RQ3 Can the isomorphism be extended to parameter orders other than the dominance (asymptotic) order?
• RQ4 How do the cellular structures of the Kazhdan–Lusztig basis and the Specht module construction compare in the specialized algebra $\mathcal{H}_{n,k}$?
• RQ5 What is the relationship between the sets $\Lambda^{\clubsuit}$ and $\Lambda^{\spadesuit}$, indexing non-zero irreducible quotients of Specht and cell modules, respectively?

Key findings

• A canonical isomorphism exists between the Specht module $\tilde{S}^{\nu}$ and the Kazhdan–Lusztig left cell module $W_{k}(\nu)$ for each bipartition $\nu$ of $n$ in the asymptotic case.
• The matrix of this isomorphism is triangular with 1s on the diagonal when both modules are equipped with standard bases ordered by the Bruhat–Chevalley order.
• The isomorphism fails for non-asymptotic monomial orders: counterexamples are constructed for the weighted lexicographic order.
• In the asymptotic case with $b > (n-1)a > 0$, the Kazhdan–Lusztig basis $\{C_w\}$ forms a cellular basis, and $W_k(\nu) \cong \tilde{S}_k^\nu$ holds.
• The sets $\Lambda^{\clubsuit}$ and $\Lambda^{\spadesuit}$, indexing non-zero irreducible quotients of Specht and cell modules, coincide precisely in the asymptotic case.
• The result generalizes the type $A_{n-1}$ isomorphism of McDonough–Pallikaros to type $B_n$, but only under the dominance order on parameters.