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Modular symmetry classes of tensors

Then we use modular symmetry classes of tensors to study the polynomial representations of GL(V ), where V is a vector space over a field of characterisitic p. At the end we introduce a non-degenerate bilinear form on a modular symmetry class. Some problems are also given. | Turk J Math 23 (1999) , 417 – 433. ¨ ITAK ˙ c TUB MODULAR SYMMETRY CLASSES OF TENSORS M. Shahryari, M. A. Shahabi Abstract We introduce the notion of modular symmetry classes of tensors and give a necessary and sufficient condition for a modular symmetry class of tensors associated with the full symmetric group to be non-zero. Then we use modular symmetry classes of tensors to study the polynomial representations of GL(V ) , where V is a vector space over a field of characterisitic p. At the end we introduce a non-degenerate bilinear form on a modular symmetry class. Some problems are also given. 1. Introduction Let V be an n-dimensional complex vector space, and G be a permutation group on m elements. Let χ be any irreducible character of G . For any g G , define the operator Pg : m O V → m O V by Pg (v1 ⊗ . . . ⊗ vm ) = vg−1 ⊗ . . . ⊗ vg−1 (m). Then the symmetry class of tensors associated with G and χ is the image of the following symmetry operator Sχ = χ(1) X χ(g)Pg . |G| g G 417 SHAHRYARI, SHAHABI The symmetry class of tensors associated to G and χ is denoted by Vχ (G). A tremendous number of articles have appeared on the past 40 years concerning symmetry classes of tensors. The two volumes book of Marvin Marcus [4] and recent book of Russell Merris [5] are valuable sources to this theme. Unfortunately there is no similar notion for the case of vector spaces over arbitrary fields in literature. The aim of this expository article is to introduce the notion of modular symmetry classes of tensors for the first time, and introduce various questions which might become the subject of further research. Let A be the field of algebraic numbers over Q and p be a prime. Let R be a valuation subring of A with the unique maximal ideal P in which P ∩ Z = pZ. Assume that F = R/P . It is well known that F is an algebraically closed field of characterisitic p. Let ? : R → F be the canonical map. Now assume that G is a permutation group on m elements and let .