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The role of bideterminants in the representation theory. (English) Zbl 1280.20049

Introduction: The purpose of this paper is to explain the role of bideterminants in the representation theory. Our goal is to present a wide scope of applications of bideterminants in the representation theory. In doing so, we will be satisfied with presenting the simplest examples whenever possible; for example instead of discussing reductive algebraic groups we will only talk about a general linear group \(\mathrm{GL}_n\). We want to provide the reader with an introduction to the topics of bideterminants instead of indulging in exhaustive technical details. The reader is advised to consult the references for further details.
In Section 1, we deal with the history and the context in which the bideterminants first emerged, namely in the invariant theory. In Section 2, we give a definition of the bideterminant and a statement of the Straightening Formula. In Section 3, we explain the role of bideterminants in the representation theory of the symmetric group. In Section 4, we review a connection of bideterminants to the representation theory of the general linear group and Schur algebra. In Section 5, we discuss the concept of bideterminants related to the representation theory of the general linear supergroup and Schur superalgebra. In Section 6, we consider the role of bideterminants in quantum groups.

MSC:

20G05 Representation theory for linear algebraic groups
17A70 Superalgebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20C30 Representations of finite symmetric groups
15A72 Vector and tensor algebra, theory of invariants
13A50 Actions of groups on commutative rings; invariant theory
05E10 Combinatorial aspects of representation theory
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