Type B Hecke Algebras

• Type B Hecke algebras are finite-dimensional associative algebras defined by generators and relations that deform the group algebra of the type B Coxeter group.
• Their representation theory is structured via cellular modules and Specht module theory using bipartitions, linking combinatorial methods with geometric invariants.
• Applications span quantum group categorification, knot invariants, and geometric representation theory through connections with quiver Grassmannians and cluster algebras.

A type B Hecke algebra is a finite-dimensional associative algebra defined by generators and relations modeling the Coxeter group of type B, or equivalently, the Weyl group of the complex simple Lie algebra of type Bₙ. Together with types A and D, these Hecke algebras control much of the modular and categorical representation theory associated to classical symmetric and reflection groups, playing a central role in the theory of quantum groups, knot invariants, and categorifications.

1. Definition and Presentation

The type B Coxeter group, $W(B_n)$, is the group of signed permutations on $n$ symbols, also realized as $\mathbb{Z}_2^n \rtimes S_n$. The Hecke algebra $H(B_n)$ is a deformation of the group algebra $\mathbb{C}[W(B_n)]$ over some ring $R$ and is generated by invertible elements $T_0,T_1,\ldots,T_{n-1}$ subject to relations encoding the Coxeter structure of $B_n$:

• Quadratic relations (deformations):

$$(T_0-u_0)(T_0+1)=0, \qquad (T_i-u)(T_i+1)=0 \quad\text{for } i=1,\ldots,n-1$$

where $u_0,u$ are parameters.

• Braid relations:

$T_0T_1T_0T_1 = T_1T_0T_1T_0$

$$T_iT_{i+1}T_i = T_{i+1}T_iT_{i+1} \quad \text{for } i>0$$

$T_iT_j = T_jT_i \quad \text{if } |i-j|>1.$

The specialization $u_0,u\to 1$ recovers the group algebra of $W(B_n)$.

2. Structural Properties and Cellular Theory

Type B Hecke algebras, as with Hecke algebras of finite Coxeter groups in general, are semisimple over sufficiently generic parameters but may be non-semisimple over roots of unity or special values. Their modular (non-semisimple) representation theory is controlled by "cellular" or "standard" modules labeled by combinatorial data generalizing the role of partitions for $S_n$ in type A. This includes bipartitions (ordered pairs of partitions), reflecting the double-coset structure $W(B_n)\cong S_n\ltimes(\mathbb{Z}/2\mathbb{Z})^n$.

The theory of "specht modules" and "cell modules" for $H(B_n)$ interacts with categorification and geometric representation theory, playing central roles in the study of canonical bases, categorifications of quantum groups, and via connections to the geometry of quiver varieties and quiver Grassmannians (Lorscheid et al., 2017).

3. Connection to Quivers and Quiver Grassmannians

The structure and representation type of type B Hecke algebras are intimately connected to the representation theory of quivers and their geometric invariants. For acyclic quivers of types A, D, and by extension B (via folding procedures and species theory), the smoothness, singularities, and cell stratification properties of quiver Grassmannians correspond precisely to the finite/tame/wild dichotomy in representation theory. Specifically:

• For types $A_n$ and $D_n$ (and hence phenomena inherited by $B_n$), all quiver Grassmannians of indecomposables are smooth and admit affine cell decompositions only in the finite type case (Lorscheid et al., 2017).
• In type B, which can also be realized via a folded type $A_{2n-1}$ or as species over valued quivers, representations have a root system and Coxeter group structure naturally matching the combinatorics of $H(B_n)$ (Lemay, 2011).
• The existence of singular or non-cellular quiver Grassmannians or those with negative Euler characteristic signals transition to wild representation type, which in the Hecke setting corresponds to modular representation-theoretic complexity.

This geometric connection underlies constructions of canonical and crystal bases, as well as links between Hecke algebra modules and cluster algebra structures (Lorscheid et al., 2017).

4. Generic and Truncated Hecke Algebras

When the Hecke algebra is taken over a field of positive characteristic or at roots of unity, it becomes non-semisimple, leading to a rich theory of "generic" and "canonical" modules analogous to the generic modules for quivers with relations (Babson et al., 2014). For truncated path algebras or finite quotients, one obtains affine charts and structure theorems for module varieties, which reflect and refine the modular representation theory of $H(B_n)$ and allow for algorithmic construction of projective resolutions and generic indecomposables beyond the finite case.

5. Applications and Categorification

Type B Hecke algebras are foundational in several advanced areas:

• Quantum Invariants and Knot Theory: The BMW algebra, a quotient of the braid group algebra refining $H(B_n)$, connects to type B knot invariants and the HOMFLY-PT polynomial, extending type A results to the framings relevant for reflection symmetries.
• Categorification and Higher Representation Theory: Type B Hecke categories appear as monoidal categories controlling categorified braid group actions, realizing the categorification of quantum groups of orthogonal type.
• Quiver Varieties and Geometric Satake Correspondence: The geometry of quiver varieties of type B and D, and their role in Springer theory and the geometry of the affine Grassmannian, is controlled by the combinatorics of the associated Hecke algebra (Lorscheid et al., 2017).
• Specht Module Theory for Bipartitions: The representation theory of $H(B_n)$ is combinatorially understood via bipartitions, with applications to the modular and symmetric function theory and connections to the crystals of higher-level Fock spaces.

6. Further Directions and Open Problems

A key open question is the extension of cellular and geometric stratification results from types A and D to exceptional types, and the full description of the cell decomposition and singularity structure for E-type quivers and Hecke algebras (Lorscheid et al., 2017). Moreover, deeper understanding of the correspondence between quiver Grassmannian invariants and representation types in type B remains an active area, especially as it connects to categorified cluster algebra theory and new geometric models for canonical bases.

There is also ongoing research in modular reduction, for example, understanding the behavior of Hecke algebras with unequal parameters (as in the more general Iwahori-Hecke algebra framework) and their interaction with the representation theory of p-adic and finite groups of Lie type B or C, as well as their connections to double affine Hecke algebras and Macdonald theory.