Francesco Regonati

The Method of Virtual Variables and Representations of Lie Superalgebras

We provide a brief account of Capelli’s method of virtual variables and of its relations with representations of general linear Lie superalgebras. More specifically, we study letterplace superalgebras regarded as bimodules under the action of superpolarization operators and exhibit complete decomposition theorems for these bimodules as well as for the operator algebras acting on them.

Whitney numbers of some geometric lattices

It is a well known fact that the Whitney numbers of a sequence of ranked posets P0, P1, P2,-.of rank 0, l, 2 .... such that every r-ranked filter of P, is isomorphic to Pr are the connection constants between the sequence of powers and the sequence of the characteristic polynomials of the posets [-Dow]. We note here that, if the posets are indeed supersolvable geometric lattices, then the sequence of characteristic polynomials is persistent in the sense of [Dam2]. Thus, in Sections 3 and 4, we will be able to transfer the two term recursion and the explicit formula for connection constants [Daml] as well as several log-concavity properties of symmetric functions [Sag2] to the Whitney numbers of those lattices. Moreover, since the roots of the characteristic polynomial of a supersolvable lattice have been given a nice combinatorial meaning [Sta2], we can also give a simple semantics for those formulas. As a consequence of our results, we obtain (Section 5) unifying proofs of several properties enjoyed by Whitney numbers of Boolean algebras, subspace lattices, partition lattices, and Dowling lattices. It turns out that these lattices form the only infinite sequences of modularly complemented geometric lattices satisfying the conditions mentioned at the beginning.

Grassmann geometric calculus, invariant theory and superalgebras

The idea of exploring and developing the deep connections between the theory of Cayley-Grassmann algebras and the invariant theory of skew-symmetric tensors was a recurrent theme of Rota’s mathematical work.

Whitney numbers of the second kind of finite modular lattices

Let L be any finite modular lattice. We prove that each interval Jz L has a symmetric and unimodal sequence of Whitney numbers of the second kind iff L may be represented as a direct product of primary q-lattices. We use the term “primary” in the classical sense of Jbnsson and Monk [7], while the term “q-lattice” is from Stanley [lo]. Actually, in the “if” part of the proof, we prove that, upon denoting by n and 1 the dimension and the rank of J, the sequence of the Whitney numbers of J is symmetric and unimodal of the form

Combinatorics and representation theory of lie superalgebras over letterplace superalgebras

We state three combinatorial lemmas on Young tableaux, and show their role in the proof of the triangularity theorem about the action of Young-Capelli symmetrizers on symmetrized bitableaux. As an application, we describe in detail the way to specialize general results to the representation theory of the symmetric group and to classical invariant theory.

Upper semimodularity of finite subgroup lattices

Abstract We prove that the subgroup lattice of a finite group is upper semimodular iff in each of its intervals [ A;B ] there are at most as many subgroups covering A as subgroups covered by B .