Let Cp denote the cyclic group of order p where p ≥ 3 is prime. We denote by V3 the indecomposable three dimensional representation of Cp over a field F of characteristic p. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants F[V3 ⊕ V3]p . Our main result confirms the conjecture of Shank[15], for this example, that all modular rings of invariants of Cp are generate...

This paper gives an algorithm for computing invariant rings of reductive groups in arbitrary characteristic. Previously, only algorithms for linearly reductive groups and for finite groups have been known. The key step is to find a separating set of invariants.

Habiro–Le invariants dominate sl2 Witten–Reshetikhin–Turaev invariants of rational homology 3–spheres at roots of unity of order coprime with the torsion. In this paper we give a formula for the Habiro–Le invariant of a rational homology 3–sphere obtained by rational surgery on a link in S. As an application, we compute this invariant for Seifert fibered spaces and for Dehn surgeries on twist k...

We formulate a notion of “geometric reductivity” in an abstract categorical setting which we refer to as adequacy. The main theorem states that the adequacy condition implies that the ring of invariants is finitely generated. This result applies to the category of modules over a bialgebra, the category of comodules over a bialgebra, and the category of quasi-coherent sheaves on an algebraic sta...

In an earlier paper [S] we have determined the structure of the linear groups over a local ring. In this note we continue the study of the classical groups over a local ring with the investigation of the orthogonal groups. Our main result (cf. Theorem 6 below) is a complete description of the invariant subgroups of an orthogonal group of noncompact type (i.e., of index ^ 1) over a local ring L ...

If G is a finite linear group over a field K such that char(K) | 6 |G|, there are various effective methods to calculate the invariant ring I of G, i.e., to find a finite system of generators of I as an algebra over K (see Sturmfels, 1993, Section 2; McShane, 1992; Kemper, 1993, 1994). These methods make use of the Reynolds operator, Molien’s formula, the Cohen–Macaulay property of invariant ri...

The two papers in this series analyze quantum invariant differential operators for quantum symmetric spaces in the maximally split case. In this paper, we complete the proof of a quantum version of Harish-Chandra’s theorem: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and a ring of Laurent polynomial invariants with resp...

We consider the system of the N vortex points with the identical strength on a sphere. As a local effect of the rotation of the sphere, we fix vortex points at the both poles, which are called the pole vortices. When the N vortex points are equally spaced along the line of latitude of the sphere, the configuration is called the “N -ring” or “N -gon”. It is a relative fixed configuration[3,4] an...

Abstract We continue with our study of the noncritical exceptional zeros Katz’ $p$-adic $L$-functions attached to a CM field $K$, following two threads. In 1st thread, we redefine (group-ring-valued) ${\mathcal {L}}$-invariant associated each ${\mathbb {Z}}_p$-extension $K_\Gamma $ $K$ in terms height pairings and interpolate them as varies universal (multivariate) group-ring-valued {L}}$-invar...