MULTIPLICITY ONE PROPERTY AND THE DECOMPOSITION OF b-FUNCTIONS

Abstract

Recently, extensive calculations have been made on b-functions of prehomogeneous vector spaces with reducible representations. By examining the results of these calculations, we observe that b-functions of a large number of reducible prehomogeneous vector spaces have decompositions which seem to be correlated to the decomposition of representations. In the present paper, we show that such phenomena can be ascribed to a certain multiplicity one property for group actions on polynomial rings. Furthermore, we give some criteria for the multiplicity one property. Our method can be applied equally to non-regular prehomogeneous vector spaces.

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

“…16 (The final "generic" case (30) involves matrices X of order 3×2n with P (X) = tr(XJX T ) 2 , which falls outside our methods since it is neither a determinant nor a pfaffian.) Finally, the most difficult result obtained in the present paper -namely, the multi-matrix rectangular Cayley identity identity (2.23) -has very recently been proven independently by Sugiyama [100, Theorem 0.1] in the context of prehomogeneous vector spaces associated to equioriented quivers of type A; his proof uses a decomposition formula found earlier by himself and F. Sato [85]. Indeed, Sugiyama proved an even more general result [100,Theorem 3.4], applying to quivers of type A with arbitrary orientation.…”

“…When applicable, this connection allows the immediate identification of a suitable operator Q(∂/∂x) -namely, the dual of P itself -and provides a general proof that the corresponding b(s) satisfies deg b = deg P and is indeed (up to a constant factor) the Bernstein-Sato polynomial of P . 6 Furthermore, this approach sometimes allows the explicit calculation of b(s) by means of microlocal calculus [59,77,78,85,86,99,100,106,109] or other methods [98]. 7 The purpose of the present paper is to give straightforward (and we hope elegant) algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new.…”

Relative invariants of 2-simple prehomogeneous vector spaces of type I

“…However, it is easily tractable and fits well to calculate the value at a given point; better than monomial expressions such as those given in [G1,U]. In particular, some of our results have already been used in [SS,Sg2,Wa] for the calculations of b-functions.…”

Singularity invariants related to Milnor fibers: survey

“…For the reducible pvs, an elementary method to calculate the b-functions of singular loci, which uses the known formula for b-functions of one variable, is presented in [124]. The decomposition formula for b-functions, which asserts that under certain conditions, the b-functions of reducible pvs have decompositions correlated to the decomposition of representations, was given in [113]. By using the decomposition formula, the b-functions of relative invariants arising from the quivers of type A have been determined in [119].…”