We describe the algebra of invariants of the vacuum module associated with an affinization of the Lie superalgebra gl(1|1). We give a formula for its Hilbert–Poincaré series in a fermionic (cancellation-free) form which turns out to coincide with the generating function of the plane partitions over the (1, 1)-hook. Our arguments are based on a super version of the Beilinson–Drinfeld–Räıs–Tauvel...

The infinite configuration space of an integrable vertex model based on Uq gl(2|2) 1 is studied at q = 0. Allowing four particular boundary conditions, the infinite configurations are mapped onto the semi-standard supertableaux of pairs of infinite border strips. By means of this map, a weight-preserving one-to-one correspondence between the infinite configurations and the normal forms of a pai...

We classify actions of groups of finite Morley rank on abelian groups ofMorley rank 2: there are essentially two, namely the natural actions of SL(V ) and GL(V ) with V a vector space of dimension 2. We also prove an identification theorem for the natural module of SL2 in the finite Morley rank category. §

We construct a generator system of the annihilator of the generalized Verma module of gl(n,C) induced from any character of any parabolic subalgebra as an analogue of minors and elementary divisors. The generator system has a quantization parameter ε and it generates the defining ideal of the conjugacy class of square matrices at the classical limit ε = 0.

The alpha-determinant unifies and interpolates the notion of the determinant and permanent. We determine the irreducible decomposition of the cyclic module of gl n (C) defined by the alphadeterminant. The degeneracy of the irreducible decomposition is determined by the content polynomial of a given partition.

In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov–Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over U (

The variety of principal minors of n× n symmetric matrices, denoted Zn, is invariant under the action of a group G ⊂ GL(2) isomorphic to (SL(2)) ×Sn. We describe an irreducible G-module of degree 4 polynomials that cuts out Zn set theoretically. This solves the set-theoretic version of a conjecture of Holtz and Sturmfels.

In a previous paper [12] with almost the same title, two of the authors considered the Lie powers L(V ) of the natural module V for GL(2,p) (where p is prime). For the case when n is not divisible by p, it was shown there that the indecomposable direct summands of L(V ) are either simple or projective, and two methods were described for calculating the relevant Krull–Schmidt multiplicities. The...

The purpose of this paper is to forge a direct link between the hit problem for the action of the Steenrod algebra A on the polynomial algebra P(n) = F2[x1, . . . , xn], over the field F2 of two elements, and semistandard Young tableaux as they apply to the modular representation theory of the general linear group GL(n,F2). The cohits Qd(n) = Pd(n)/Pd(n)∩A+(P(n)) form a modular representation o...