Seminormal forms and Gram determinants for cellular algebras

Abstract

ABSTRACT. This paper develops an abstract framework for constructing "seminormal forms" for cellular algebras. That is, given a cellular R-algebra A which is equipped with a family of JM-elements we give a general technique for constructing orthogonal bases for A, and for all of its irreducible representations, when the JM-elements separate A. The seminormal forms for A are defined over the field of fractions of R. Significantly, we show that the Gram determinant of each irreducible A-module is equal to a product of certain structure constants coming from the seminormal basis of A. In the non-separated case we use our seminormal forms to give an explicit basis for a block decomposition of A.

Categorification of Wedderburn’s basis for $${\mathbb{C}}[S_{n}]$$

“…In particular, our setup is completely different from the combinatorial approach of [21]. There are alternative combinatorial approaches to the construction of a basis for C[S n ] and some related algebras in which the regular representation decomposes into a direct sum of irreducibles, see [22], [19], [20], [15], [16]. There are also alternative combinatorial constructions (e.g.…”

Cellular Bases for Algebras with a Jones Basic Construction

“…Several papers in the literature have aimed at generalizations or axiomatizations of the Murphy basis, the seminormal basis, and the set of Jucys-Murphy elements, for example [9,14,29,37]. The present paper is also a contribution to this theme.…”

“…It follows from [14,Propositions 3.6 and 3.7] that the Jucys-Murphy elements act triangularly on our Murphy bases. Hence, Mathas' theory of cellular algebras with Jucys-Murphy elements and seminormal representations [29] can be applied. It is shown in [1] that triangularity actually holds with respect to dominance order, strengthening the triangularity statements of [13].…”

The Representations of Cyclotomic BMW Algebras, II

“…By [20, 4.5], we have s = t if and only if c s (k) = c t (k), 1 ≤ k ≤ n. In other words, the separate condition in the sense of [15, 2.8] holds for B r,n over F. Although the results in [15] are stated for cellular algebras, they are still available for cellular algebras in weak version. So, we can use standard arguments in [15] to construct an orthogonal basis for ( f, λ) as follows.…”

“…Following [18], we shall construct a B r,n−1 -filtration for M. Via it, we shall construct an R-basis for M, called the JM-basis in the sense of [15]. This enables us to use standard arguments in [15] to construct an orthogonal basis for M under so called separate condition in the sense of [15]. The key is that the Gram determinants associated to M which are defined by the JM-basis and the previous orthogonal basis are the same.…”