A note on multiplicity-free tensor representations. (English) Zbl 0835.20058
Let \(V\) be a finite-dimensional vector space over a field \(F\) of characteristic 0. Let \(T\) be the tensor algebra of \(V\), and \(G = GL(V)\). Then \(T\) is in a natural way an \(FG\)-module. It is well-known that \(T\) is completely reducible, and each polynomial \(FG\)-module occurs (with large multiplicity) as an irreducible constituent of \(T\).
Let \(A\) denote the algebra (associative with 1) generated by \(V\) subject to the sole condition that the squares of the elements of \(V\) lie in the centre of \(A\). Then \(A\) is both a quotient algebra and a quotient \(FG\)- module of \(T\). In the paper under review the author proves that the \(FG\)- module \(A\) is isomorphic to the direct sum of the irreducible polynomial \(FG\)-modules (Theorem 1). In other words, \(A\) is an explicit model for the polynomial \(F\)-representations of \(GL(V)\). Let \(Z\) be the subalgebra of \(A\) generated by the squares of the elements of \(V\). The author proves that \(A\) is a free \(Z\)-module of finite rank and gives a nice combinatorial description of a basis of a free \(Z\)-module \(A\) (Proposition 6).
Let \(A\) denote the algebra (associative with 1) generated by \(V\) subject to the sole condition that the squares of the elements of \(V\) lie in the centre of \(A\). Then \(A\) is both a quotient algebra and a quotient \(FG\)- module of \(T\). In the paper under review the author proves that the \(FG\)- module \(A\) is isomorphic to the direct sum of the irreducible polynomial \(FG\)-modules (Theorem 1). In other words, \(A\) is an explicit model for the polynomial \(F\)-representations of \(GL(V)\). Let \(Z\) be the subalgebra of \(A\) generated by the squares of the elements of \(V\). The author proves that \(A\) is a free \(Z\)-module of finite rank and gives a nice combinatorial description of a basis of a free \(Z\)-module \(A\) (Proposition 6).
Reviewer: A.A.Premet (Manchester)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20C30 | Representations of finite symmetric groups |
| 20G15 | Linear algebraic groups over arbitrary fields |
Keywords:
finite-dimensional vector space; tensor algebra; irreducible constituent; irreducible polynomial \(FG\)-modules; polynomial \(F\)-representations; free \(Z\)-module of finite rankReferences:
| [1] | Green, J. A., Polynomial Representations of \(GL_n\), (Lecture Notes in Mathematics, Vol. 830 (1980), Springer: Springer Berlin) · Zbl 0453.20029 |
| [2] | Inglis, N. F.J.; Richardson, R. W.; Saxl, J., An explicit model for the complex representations of \(S_n\), Arch. Math., 54, 258-259 (1990) · Zbl 0695.20008 |
| [3] | Kljac̆ko, A. A., Models for complex representations of the groups GL(n,q) and Weyl groups, Soviet Math. Dokl., 24, 496-499 (1981) · Zbl 0496.20031 |
| [4] | Macdonald, I. G., Symmetric Functions and Hall Polynomials (1979), Oxford University Press: Oxford University Press Oxford · Zbl 0487.20007 |
| [5] | Razmyslov, Yu. P., On Engel Lie algebras, Algebra i Logika, 10, 33-44 (1971), (in Russian) |
| [6] | Verma, D.-N., On a classical stability result on invariants of isotypical modules, J. Algebra, 63, 15-40 (1980) · Zbl 0444.14011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
