arXiv subjects

T. Geetha

Publications and source records attributed to T. Geetha.

5 recordsLinked to original sources

arXiv2020

On the determinant of representations of generalized symmetric groups

In this paper we study the determinant of irreducible representations of the generalized symmetric groups $\mathbb{Z}_r \wr S_n$. We give an explicit formula to compute the determinant of an irreducible representation of $\mathbb{Z}_r \wr S_n$. Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, for given integer $n$, $r$ an odd prime and $\zeta$ a nontrivial multiplicative character of $\mathbb{Z}_r \wr S_n$ with $n<r$, we obtain an explicit formula to compute $N_{\zeta}(n)$, the number of irreducible representations of $\mathbb{Z}_r \wr S_n$ whose determinant is $\zeta$.

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arXiv2016

Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups

Young's orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a symmetric group. In this article, we describe a recursive algorithm to write down the expansion of each Gelfand-Tsetlin basis vector for an irreducible representation of an alternating group in terms of Young's orthogonal basis of the ambient representation of the symmetric group. This algorithm is implemented with the Sage Mathematical Software.

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arXiv2014

Graphic Interpretation of the Structure Constants of the Schur Algebra

Issai Schur, in his doctoral thesis (1901) introduced the Schur algebra to study the polynomial representation theory of the general linear group. He described a basis of this algebra and structure constants. Later, Miguel Mendez (2001) gave a graph-theoretic description of Schur's basis and computed the structure constants. In this presentation, we will give a new graphic interpretation of the basis of Schur algebra and use it to give a description of the structure constants which is equivalent to the one given by Mendez.

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arXiv2013

Centre of the Schur Algebra

We describe a basis of the centre of the Schur algebra which comes from conjugacy classes in the symmetric group via Schur-Weyl duality. We give a combinatorial description of expansions of these basis elements in terms of the basis originally used by Schur. The primitive central idempotents of the Schur algebra can be written down using this basis and the character table of the symmetric group. Along the way we prove a result on the non-singularity of the submatrix of the character table matrix of a symmetric group obtained by taking rows and columns indexed by partitions with at most n parts for any n.

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arXiv2012

Cellularity of Wreath Product Algebras and $A$--Brauer algebras

A cellular algebra is called cyclic cellular if all cell modules are cyclic. Most important examples of cellular algebras appearing in representation theory are in fact cyclic cellular. We prove that if $A$ is a cyclic cellular algebra, then the wreath product of $A$ with the symmetric group on $n$ letters is also cyclic cellular. We also introduce $A$--Brauer algebras, for algebras $A$ with an involution and trace. This class of algebras includes, in particular, $G$--Brauer algebras for non-abelian groups $G$. We prove that if $A$ is cyclic cellular then the $A$--Brauer algebras $D_n(A)$ are also cyclic cellular.

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