Showing papers on "Symplectic vector space published in 1971"

The Dimensions of Irreducible Tensor Representations of the Orthogonal and Symplectic Groups

Abstract: It is well known [13] that the irreducible tensor representations (IRs) of the unitary, orthogonal, and symplectic groups in an n-dimensional space may be specified by means of Young tableaux associated with partitions (σ)s = (σ 1, σ 2, …, σp ) with σ 1 + σ 2 + … + σp = s. Formulae for the dimensions of the corresponding representations have been established [1; 8; 9; 13] in terms of the row lengths of these tableaux. It has been shown [12] for the unitary group, U(n), that the formula may be written as a quotient whose numerator is a polynomial in n containing s factors, and whose denominator is a number independent of n, which likewise may be expressed as a product of s factors. This formula is valid for all n.

Symplectic geometries over finite abelian groups

Abstract: In the paper one investigates symplectic abelian groups that are the group-theoretical analog of symplectic linear spaces. Bibliography: 3 items.

Divisibility conditions on characteristic numbers of stably symplectic manifolds

Abstract: It is shown that every cohomology characteristic number of an 8k+4 [resp. 16k+8] dimensional stably symplectic manifold is divisible by 4 [resp. 2] and that certain characteristic numbers of 2-dimensional stably symplectic manifolds are divisible by 2 and 4. The proofs depend on symplectic cobordism operations. Using explicit manifold constructions of Stong [5 ] it is shown that these results are to a large extent the best possible. R. Stong [5] described certain "Riemann-Roch" relations on the cohomology characteristic numbers of stably symplectic manifolds. In a previous paper [4] we used these relations to compute certain differentials in the Adams spectral sequence for the 2-primary stable homotopy of MSp. Stong's relations give rise to extremely tedious calculations which become prohibitive after the 24-stem and which, in any case, do not give any simple results which are valid in an infinite family of stems. In this note we show that results of P. S. Landweber [2] and simple arithmetic yield characteristic number relations relevant to the Adams spectral sequence in arbitrarily high stems. All homology and cohomology will be with integral coefficients except as otherwise noted. THEOREM 1. Let f: S4k"-*MSp be a stable map and let S4k generate H4k(S4k). Then (i) If k 1(2), f*(S4k) is divisible by 4. (ii) If k 2(4), f*(s4k) is divisible by 2. COROLLARY 2. Every cohomology characteristic number of a 4kdimensional stably symplectic manifold is divisible by 4 [resp. 2], if k=1(2) [resp. k=2(4)]. We will prove part (i) of Theorem 1; part (ii) may be proved by similar methods or obtained as a consequence of a theorem of E. E. Received by the editors March 23, 1970. AMS 1968 subject classifications. Primary 5710, 5732.