yψ ց m//G a/WV → t /WG where a ⊆ t is a subspace, WV ⊆ GL(a ) is a reflection group, and ψ is faithfully flat with irreducible generic fibers. Note that WV being a reflection group implies that a /WV is isomorphic to an affine space. The group WV is a very subtle invariant of V which plays a similar role as the Weyl group of the restricted root system for a symmetric space. The morphism ψ facto...

The adjoint representation of a simple Lie algebra g admits a deformation into an irreducible representation R of the quantum group Uq(g). In this paper for a simply-laced g we realize R as the Grothendieck group of a particular abelian category C. There are exact functors from C to C which on the Grothendieck group act as the quantum group generators Eα, Fα, where α varies over simple roots. V...

Let G ⊂ SL(n, C) be a finite subgroup and φ:Y → X = C/G any resolution of singularities of the quotient space. We prove that crepant exceptional prime divisors of Y correspond one-to-one with “junior” conjugacy classes of G. When n = 2 this is a version of the McKay correspondence (with irreducible representations of G replaced by conjugacy classes). In the case n = 3, a resolution with KY = 0 ...

In a previous paper (Groups, Geometry, and Dynamics, 2015), we introduced a new homological invariant ε for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite group G with a fixed homological invariant ε, if the genus g′ of the quotient curve satisfies g′ 0, is irreducible (and non-empty if and only...

Consider a linear multiplicity free action by a compact Lie group K on a finite dimensional hermitian vector space V . Letting K act on the associated Heisenberg group HV = V × R yields a Gelfand pair. In previous work we have applied the Orbit Method to produce an injective mapping Ψ from the space ∆(K,HV ) of bounded K-spherical functions on HV to the space h ∗ V /K of K-orbits in the dual of...

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We define Γ(G) to be the graph whose vertex set is cd(G) − {1}, and there is an edge between a and b if (a, b) > 1. We prove that if Γ(G) is a complete graph, then G is a solvable group.

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let W = V ⊕ Fe with the form Q extending q with Q(e) = 1. Consider the standard embedding O(V ) ↪→ O(W ) and the two-sided action of O(V )×O(V ) on O(W ). In this note we show that any O(V )×O(V )-invariant distribution on O(W ) is invariant with respect to transposition. This result was...

Hermann proposed that mesomorphic media should be classified by assigning certain "statistical symmetry groups" to each possible partially ordered array. Two translational groups introduced were called superordinate and subordinate. We find that the average density in such a partially ordered medium has the superordinate symmetry ~, while the pair correlation function has the subordinate symmet...

We review and compare two definitions of rough set approximations. One is defined by a pair of sets in the universe and the other by a pair of sets in the quotient universe. The latter definition, although less studied, is semantically superior for interpreting rule induction and is closely related to granularity switching in granular computing. Numerical measures about the accuracy and quality...

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H,G,α, β) is deformed using a combinatorial datum (σ, v, r) consisting of an automorphism σ of H , a permutation v of the set G and a transition map r : G → H in order to obtain a new matched pair `