Abhyankar, Shreeram S. ; Ghorpade, Sudhir R.
(1991)
*Young tableaux and linear independence of standard monomials in multiminors of a multimatrix*
Discrete Mathematics, 96
(1).
pp. 1-32.
ISSN 0012-365X

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/001236...

Related URL: http://dx.doi.org/10.1016/0012-365X(91)90467-G

## Abstract

As a culmination of the efforts of the invariant theorists from Clebsch, Gordan, Young, to Rota, in 1972 Doublet-Rota-Stein proved the Straightening Law which says that the standard monomials in the minors of a matrix X, which correspond to standard bitableaux, form a vector space basis of the polynomials ring K[X] in the indeterminate entries X over the coefficient field K. Now we may ask what happens to this when we consider 'higher dimensional' matrices by using cubical, 4-way,...,q-way determinants which were already introduced by Cayley in 1843. In the present paper we show that, for every q >2, the standard monomials in the multiminors of the multimatrix X are linearly independent over K. In a forthcoming paper it will be shown that they do not span the polynomial ring K[X]. The proof of linear independence given in this paper also applies to the classical case of q=2.

Item Type: | Article |
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ID Code: | 118 |

Deposited On: | 17 Sep 2010 06:47 |

Last Modified: | 10 May 2011 08:28 |

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