arXiv subjects
Publications and source records attributed to Gerald Cliff.
We show that the Weil representation of the symplectic group Sp(2n,F), where F is a non-archimedian local field, can be realized over the field obtained from the rationals by adjoining the square roots of p and -p, where p is the residue characteristic of F; p is assumed to be odd.
We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n,K), where K is an infinite field of characteristic not 2. The bideterminants are indexed by pairs of Young tableaux which are O(n)-standard in the sense of King-Welsh. We also give an explicit filtration of K[O(n)] as an O(n,K)-bimodule, whose factors are isormorphic to the tensor product of orthogonal analogues of left and right Schur modules.
Let B be the crystal basis of the minus part of the quantized enveloping algebra of a semi-simple Lie algebra. Kashiwara has shown that B has a combinatorial description in terms of an embedding of B into the tensor product of B and k abstract crystals B_{i_j}, j=1,2,...,k, where the longest word in the Weyl group is s_{i_1}...s_{i_k}. We give an explicit description of the image of this embedding for classical Lie algebras of types A, B, C, D. This description is in terms of semi-standard Young tableaux of types A, B, C, D defined by Kashiwara and Nakashima.