Seminormal forms and Gram determinants for cellular algebras

By Andrew Mathas, M. Soriano. 2006

85 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e5cb19f58936d609aee653b13cafe6154d05a73d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Homogeneous representations of Khovanov-Lauda algebras (2008, methodology): “A module M ∈ R�-mod is called calibrated if y1,...,yd act as zero on M. Other authors might use different terminology here, for example Gelfand-Zetlin [Ch, OV], completely splittable [K1, K2, Ru], skew [Mo], seminormal [ Ma ], etc.”
Representation theory of symmetric groups and related Hecke algebras (2009, unknown): “[156, 38, 151].”
Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A (2009, background): “The degenerate cyclotomic Hecke algebras fit into the general framework considered in [30], so e(i)O ∈ HO n by [30, Lemma 4.”