Hall Polynomials for Symplectic Groups, II

AbstractThis is the second paper on Hall polynomials for symplectic groups. The definition is analogous to that of Hall polynomials for general linear groups. In both papers we compute the number of all totally isotropic subspacesWof type μ in a vector space with symplectic geometryVof type λ denotedgλμ(see Section 0.1 for definitions). Let the dimensions ofVandWbe 2mandm, respectively. We represent the basis ofVof type λ=rd11rd22…rdsswith a diagram consisting ofsblocks. The entries of this diagram represent elements of a basis ofVconsisting ofmhyperbolic pairs (see Definition 0.1.1). Because of the existence of S–S basis (see 0.2 for definition) we always have an obvious totally isotropic subspace. Our method is to “deform” this initial subspace into all others by adding and discarding basis elements from the current basis. We must keep track of the type of subspace we thus obtain and the geometric structure (that is, the positions of hyperbolic pairs in the diagram) of the entire space, ensuring isotropy at each step. In our first paper [10] we obtain the result for the case when λ=rd, corresponding to a diagram with one block. In this case we can choose our basis so that each type ofWcorresponds to one geometric structure ofVonly. This makes our task and notation simpler and we obtain a closed formula forgλμ(see Theorem 0.5.5). Unfortunately it is not so in the case of λ=rd11rd22…rdss,s≠1, where two subspaces may have the same type but induce different geometric structures inV(see the example in Section 1). This case is studied below. The Hall polynomials for general linear groups have been computed by T. Klein [4] (later, I. G. Macdonald [5] and F. M. Maley [6]). Our method differs from that of either author substantially. Many of the results of [10] are used in this paper. A quick review of those results is given in the first section of this paper