Narrow Results

A new basis of the $q$-Brauer algebra is introduced, which is a lift of Murphy bases of Hecke algebras of symmetric groups. This basis is shown to be a cellular basis in the sense of Graham and Lehrer. Using combinatorial tools we prove that the non-isomorphic simple $q$-Brauer modules are indexed by the $e(q^2)$-restricted partitions of $n-2k$ where $k$ is an integer, $0\le k\le [n/2]$. When the $q$-Brauer algebra has low dimension a criterion of semisimplicity is given, which is used to show that the $q$-Brauer algebra is in general not isomorphic to the BMW-algebra.

We provide a diagrammatic approach to compute the algebraic structures on the Grothendieck group of a tower of the Temperley–Lieb algebras.

In the present paper we describe a class of Φ-Auslander-Yoneda algebras over K[x]/(xⁿ) in terms of quivers with relations, and prove that they are actually cellular algebras in the sense of Graham and Lehrer.

Let $R$ be a ring with an automorphism $𝜑$ of order two. We introduce the definition of $𝜑$-centrosymmetric matrices. Denote by $M_n(R)$ the ring of all $n\times n$ matrices over $R$, and by $S_n(𝜑,R)$ the set of all $𝜑$-centrosymmetric $n\times n$ matrices over $R$ for any positive integer $n$. We show that $S_n(𝜑,R)\subseteq M_n(R)$ is a separable Frobenius extension. If $R$ is commutative, then $S_n(𝜑,R)$ is a cellular algebra over the invariant subring $R^{𝜑}$ of $R$.

In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a sequence of walled Brauer algebras. This approach also allows us to calculate the structure constants of multiplication over the Grothendieck ring of the sequence.