Let κ be an U-invariant reproducing kernel and let H (κ) denote the reproducing kernel Hilbert C[z1, . . . , zd]-module associated with the kernel κ. Let Mz denote the d-tuple of multiplication operators Mz1 , . . . ,Mzd on H (κ). For a positive integer ν and d-tuple T = (T1, . . . , Td), consider the defect operator

The Hilbert series and degree bounds play signiicant roles in computational invariant theory. In the modular case, neither of these tools is available in general. In this article three results are obtained, which provide partial remedies for these shortcomings. First, it is shown that the so-called extended Hilbert series, which can always be calculated by a Molien type formula, yields strong c...

Let κ be an U-invariant reproducing kernel and let H (κ) denote the reproducing kernel Hilbert C[z1, . . . , zd]-module associated with the kernel κ. Let Mz denote the d-tuple of multiplication operators Mz1 , . . . ,Mzd on H (κ). For a positive integer ν and d-tuple T = (T1, . . . , Td), consider the defect operator

Each matrix representation P: G —> GLn(K) of a finite Group G over a field K induces an action of G on the module A over the polynomial algebra A = K [ x 1 , . . . , xn]. The graded A-submodule M(P) of A generated by the orbit of (x1, ..., xn) is studied. A decomposition of MO) into generic modules is given. Relations between the numerical invariants of P and those of M(P), the latter being eff...

We will show that Feichtinger’s algebra S0(R ) (also known as modulation space M 0 (R ) and well defined for general locally compact Abelian groups) is a natural Hilbert module for quantum tori and that a result by V. Losert can be interpreted as implying that the construction of such Hilbert modules gives Feichtinger’s algebra, if one considers the category of time-frequency invariant homogeno...

For each integer n ≥ 2, we demonstrate that a 2n-dimensional generalized MICZ-Kepler problem with magmatic charge μ = 0 or 1/2 has an Spin(2, 2n+ 1) dynamical symmetry which extends the manifest Spin(2n) symmetry. The Hilbert space of bound states is shown to form a unitary highest weight Spin(2, 2n+1)-module which occurs at the first reduction point in the Enright-Howe-Wallach classification d...

The idea of associating a free resolution to a finitely generated module was introduced in two famous papers by Hilbert in 1890 [Hi1] and 1893 [Hi2]. He proved Hilbert’s Syzygy Theorem 4.9, which says that the minimal free resolution of every finitely generated graded module over a polynomial ring is finite. Since then, there has been a lot of progress on the structure and properties of finite ...

Lerch’s formulae for Euler quotients in the rings Z and Fq[t] have already been studied. In this paper, we extend the study of these quotients to number fields and the Carlitz module. In the number fields case, we prove a version of Lerch’s formula for OKHil , the ring of integers of the Hilbert class field of a number field K. In the Fq[t] case, we replace the usual multiplication in Fq[t] wit...

For each integer n ≥ 2, we demonstrate that a 2n-dimensional generalized MICZ-Kepler problem has an g Spin(2, 2n+1) dynamical symmetry which extends the manifest Spin(2n) symmetry. The Hilbert space of bound states is shown to form a unitary highest weight g Spin(2, 2n+1)-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest ...

Let Hc be the rational Cherednik algebra of type An−1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators with associated graded ring grUc = C[h ⊕ h∗]W , where W is the n-th symmetric group. Using the Z-algebra construction from [GS] it is also possible to associate to a filtered Hcor Uc-module M a coherent sheaf Φ̂(M) on the Hilbert scheme Hilb(n). Using ...