Notes on Polynomial Representations of General Linear Groups
نویسنده
چکیده
Let F be an infinite field. For each pair (i, j) with 1 ≤ i, j ≤ n, let Xij : GLn(F ) → F be the coordinate function sending a matrix x to its entry xij . Let F [Xij ] denote the algebra generated by these functions. As this notation suggests, we identify F [Xij ] with the polynomial ring in n 2 indeterminants; this is admissible because the field F is infinite. Let V be a finite-dimensional F -representation of GLn(F ). We say that V is a polynomial representation if there is a basis v1, . . . , vd of V such that the functions fab : GLn(F )→ F for 1 ≤ a, b ≤ d defined by
منابع مشابه
Notes on representations of finite groups
1 Basic notions 4 1.1 ⊕, ⊗ and Hom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Invariant vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Invariant subspaces and irreducible representations . . . . . . . . . . . . . . . . . . . 5 1.4 The group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
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