Roberto La Scala

COSTANDARD MODULES OVER SCHUR SUPERALGEBRAS IN CHARACTERISTIC p

In this paper we consider the problem of describing the costandard modules ∇(λ) of a Schur superalgebra S(m|n,r) over a base field K of arbitrary characteristic. Precisely, if G = GL(m|n) is a general linear supergroup and Dist(G) its distribution superalgebra we compute the images of the Kostant ℤ-form under the epimorphism Dist(G) → S(m|n,r). Then, we describe ∇(λ) as the null-space of some set of superderivations and we obtain an isomorphism ∇(λ) ≈ ∇(λ+|0) ⊗ ∇(0|λ-) assuming that λ = (λ+|λ-) and λm = 0. If char(K) = p we give a Frobenius isomorphism ∇(0|pμ) ≈ ∇(μ)p where ∇(μ) is a costandard module of the ordinary Schur algebra S(n,r). Finally we provide a characteristic free linear basis for ∇(λ|0) which is parametrized by a set of superstandard tableaux.

SUPER RSK-ALGORITHMS AND SUPER PLACTIC MONOID

We construct the analog of the plactic monoid for the super semistandard Young tableaux over a signed alphabet. This is done by developing a generalization of the Knuths relations. Moreover we get generalizations of Greenes invariants and Young–Pieri rule. A generalization of the symmetry theorem in the signed case is also obtained. Except for this last result, all the other results are proved without restrictions on the orderings of the alphabets.

Gröbner bases of ideals invariant under endomorphisms

Abstract We introduce the notion of Grobner S -basis of an ideal of the free associative algebra K 〈 X 〉 over a field K invariant under the action of a semigroup S of endomorphisms of the algebra. We calculate the Grobner S -bases of the ideal corresponding to the universal enveloping algebra of the free nilpotent of class 2 Lie algebra and of the T-ideal generated by the polynomial identity [ x , y , z ] = 0 , with respect to suitable semigroups S . In the latter case, if | X | > 2 , the ordinary Grobner basis is infinite and our Grobner S -basis is finite. We obtain also explicit minimal Grobner bases of these ideals.

DEFINING RELATIONS OF LOW DEGREE OF INVARIANTS OF TWO 4 × 4 MATRICES

The trace algebra Cnd over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n, d ≥ 2. Minimal sets of generators of Cnd are known for n = 2 and 3 for any d and for n = 4 and 5 and d = 2. The explicit defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Defining relations of minimal degree for n = 3 and any d are also known. The minimal degree of the defining relations of any homogeneous minimal generating set of C42 is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree ≤ 14. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple.

An algorithm for complexes

For computing free resolutions over a polynomial ring the usual approach consists in iterating the Buchburgers algorithm for each module in the resolution. In this paper, we propose one single algorithm which can be viewed as a generalization of Buchbergers to chain complexes. The algorithm is based on the use of syzygies, due to Mo¨ller, Mora and Traverso, as criteria for avoiding useless computation of S-polynomials. Some strategies for the pairs selection in complexes are studied and tested in some experiments.

Monomial right ideals and the Hilbert series of noncommutative modules

In this paper we present a procedure for computing the rational sum of the Hilbert series of a finitely generated monomial right module

Algebraic construction and numerical behavior of a new s-consistent difference scheme for the 2D Navier–Stokes equations

N

Robinson–Schensted–Knuth Correspondence and Weak Polynomial Identities of M1,1(E)

over the free associative algebra

ACTION OF THE BOREL GROUP ON MONOMIAL IDEALS

K\langle x_1,\ldots,x_n \rangle

A Koszul Decomposition for the Computation of Linear Syzygies

. We show that such procedure terminates, that is, the rational sum exists, when all the cyclic submodules decomposing