Kazhdan-type Hecke-Isomorphisms

• Kazhdan-type Hecke-algebra isomorphisms are explicit structural correspondences among Hecke algebras that generalize Kazhdan–Lusztig constructions.
• They leverage combinatorial, module-theoretic, and geometric techniques to match cellular bases and W-graph structures across various algebraic models.
• These isomorphisms enable efficient recursive algorithms for basis computations and underpin deep connections in representation theory and categorification.

A Kazhdan-type Hecke-algebra isomorphism is a class of structural correspondences and explicit isomorphisms between Hecke algebras—often governed by geometric, combinatorial, or module-theoretic data—that generalize the archetypal constructions of Kazhdan, Lusztig, and collaborators. Such isomorphisms typically relate (parabolic, affine, graded, cyclotomic, diagrammatically enhanced, or modular) Hecke algebras to various explicit algebraic or combinatorial models and often underlie deep relations between representation theory, categorification, and geometry.

1. Foundations: Kazhdan–Lusztig Theory, Module Bases, and Cellular Structures

At the foundation of Kazhdan-type isomorphisms is the Kazhdan–Lusztig construction of special bases for Hecke algebras associated with finite or affine Coxeter groups. For a finite Coxeter group $W$ with generators $S$, the one-parameter Hecke algebra $\mathcal{H} = \mathcal{H}(W)$ over $A = \mathbb{Z}[q, q^{-1}]$ has standard basis $\{T_w\}$ and relations $(T_s + q^{-1})(T_s - q) = 0$ for $s \in S$. The Kazhdan–Lusztig (KL) basis $\{C_w\}$ is constructed recursively via:

$$C_w = T_w + \sum_{y < w} P_{y,w} T_y, \qquad P_{y,w} \in qA^+,$$

with $P_{y,w}$ (the KL polynomials) encoding key combinatorial data. This basis is characterized by invariance under the bar-involution $q \leftrightarrow q^{-1}$ and triangularity relative to the standard basis.

This recursive structure can be localized to submodules or quotient modules such as the generic Specht modules $$S^J = (\mathcal{H}^{\leq w_J} + \mathcal{H}^J)/\mathcal{H}^J$$, where $\mathcal{H}^{\leq w_J}$ is based on elements less than or equal to the longest element $w_J$ of a parabolic subgroup $W_J$. The existence of a cellular basis—an algebra basis compatible with an anti-involution and encoding a natural filtration—entrenches the module structure and supports algorithmic computations, such as the construction of $W$-graphs that capture the module's combinatorial symmetries (Yin, 2010).

2. Parabolic Subgroups, Cells, and the Significance of the Longest Element

The identification and use of coset representatives, notably the distinguished maximal representatives $E_J$ associated to parabolic subgroups $W_J$, play a central role in these isomorphism frameworks. In particular, the cell (or union of cells) containing $w_J$ governs the structure of the associated Specht module, yielding bases and decompositions that are stable under natural group-theoretic actions.

The importance of $w_J$ results from its selection of a left cell (or union of Kazhdan–Lusztig left cells) whose algebraic and combinatorial invariants are then inherited by the module. This leads to a natural identification of cell modules (such as $S^J$) with Specht modules, and enables an explicit construction of the KL basis within these modules via the recursive expansion:

$C_w = T_w + \sum_{y < w,\, y \in E_J} P_{y,w} T_y,$

where the relative KL polynomials $P_{y,w}$ satisfy strict vanishing, normalization, and degree conditions.

Such cellular decompositions allow comparison between module categories of different Hecke algebras—once the combinatorial structures are matched, representation-theoretic properties and isomorphisms follow.

3. W-Graphs, Cellular Bases, and Algorithms

A haLLMark of the KL theory is the definition of combinatorial $W$-graphs, structures whose vertices correspond to basis elements (typically $C_w$ for $w \in E_J$), with descent set functions $I(w) = \{s\in S: l(sw) < l(w)\}$ and edge weights (often defined as the coefficient of $q$ in $-P_{y,w}$) reflecting the recursive expansion coefficients. The $W$-graph encapsulates the action of the Hecke algebra's generators, as captured in relations such as:

$$T_s C_w = q C_w + C_{sw} + \sum_{z < w} \mu(z,w) C_z,$$

which recursively define the module structure and encode the combinatorial complexity of KL theory in a tractable, graph-theoretical framework.

In type $A$ (symmetric group case), this structure is completely combinatorial: $J$ indexes Young subgroups, $E_J$ bijects with standard tableaux, and the cellular/Murphy bases are shown to transition exactly into the $W$-graph basis via explicitly recursive formulas, unifying tableau-theoretic and KL-theoretic representation bases.

4. Hecke-Algebra Isomorphisms: Matching Cellular Structures

Kazhdan-type Hecke-algebra isomorphisms arise as a consequence of the tight alignment between cellular bases and module filtrations. When two Hecke algebras—possibly corresponding to different group-theoretic contexts, parameter specializations, or constructions—exhibit equivalent cellular or $W$-graph structures, one can often construct explicit algebra isomorphisms that match their module categories.

Formally, this isomorphism is grounded in the identification:

Cellular Basis	Indexing Set $E_J$	Recursive Define in Terms of
$\{C_w\}$	$E_J$	$T_w$ and $P_{y,w}$
Murphy Basis	Standard Tableaux	$m_{st} = T_{d(s)} C_{w_J} T_{d(t)}^*$

Concretely, when two module categories (for different Hecke algebras) have isomorphic $W$-graph structures (identical vertices, descent maps, and edge weights), the isomorphism of their algebras is detected at the level of their representation theory, and a “Kazhdan-type” isomorphism is realized.

This correspondence not only relates modules but provides a “coordinate system” for explicit comparison, computation, and often for deducing equivalences between categories arising in seemingly disparate settings—such as between the cyclotomic quotients of Hecke algebras and quiver Hecke (KLR) algebras, or between algebras related to distinct types under a functorial framework (Yin, 2010).

5. Algorithmic and Computational Implications

A practical consequence of these isomorphisms is the availability of explicit recursive algorithms for constructing bases, determining structure constants, and computing the action of Hecke algebra generators. The recursive nature of the $P_{y,w}$ and their inversion via $Q_{y,w}$ allows for efficient, stepwise reduction of representation-theoretic problems to computations in cellular and $W$-graph bases.

Moreover, in type $A$, explicit connections between Murphy's basis and the KL basis enable direct calculation in terms of familiar combinatorics—partitioning the module as cellular and controlling the construction via tableau combinatorics.

In settings where Murphy's basis coincides with the $W$-graph basis (as is the case for $\mathfrak{S}_n$), computational implementations of module operations, character computations, and homogeneity checks are all streamlined.

6. Broader Significance and Interrelations

Kazhdan-type Hecke-algebra isomorphisms are fundamental to the synthesis of combinatorial, algebraic, and geometric representation theory. They serve as the bridge connecting canonical bases (Kazhdan–Lusztig, Murphy, $W$-graphs), uncovering deep combinatorial invariants (cells, tableaux, descent sets), and structuring module categories in a manner amenable to explicit isomorphism construction.

These results do not merely clarify the structure of Hecke algebras and their modules, but enable transfer of results and techniques across types, group settings, and even to categorical frameworks (e.g., $W$-graph algebras, categorifications, and equivalences with diagrammatic and geometric realizations). Examples in type $A$ serve as templates for further generalizations to affine, graded, and cyclotomic settings, where the core mechanism—the matching of cellular or $W$-graph structures—remains central to the formulation of isomorphisms and equivalences.

As such, Kazhdan-type isomorphisms underpin much of the modern perspective on the representation theory of Hecke algebras and its generalizations, with far-reaching implications in combinatorics, geometry, and the theory of categorification.