Minimal logarithmic signatures for finite groups of Lie type

A logarithmic signature (LS) for a finite group G is an ordered tuple α =  [A ₁, A ₂, . . . , A _n ] of subsets A _i of G, such that every element ${g \in G}$ can be expressed uniquely as a product g =  a ₁ a ₂ . . . a _n , where ${a_i \in A_i}$ . The length of an LS α is defined to be ${l(\alpha)= \sum^{n}_{i=1}|A_i|}$ . It can be easily seen that for a group G of order ${\prod^k_{j=1}{p_j}^{m_j}}$ , the length of any LS α for G, satisfies, ${l(\alpha) \geq \sum^k_{j=1}m_jp_j}$ . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS) (González Vasco et al., Tatra Mt. Math. Publ. 25:2337, 2002). The MLS conjecture states that every finite simple group has an MLS. This paper addresses the MLS conjecture for classical groups of Lie type and is shown to be true for the families PSL _n (q) and PSp _2n (q). Our methods use Singer subgroups and the Levi decomposition of parabolic subgroups for these groups.     

Minimal permutation representations of finite simple classical groups. Special linear,symplectic, and unitary groups

This work was supported by the Russian Foundation for Fundamental Research, grant 93-011-1501.     

Homological stability for classical groups

Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi subgroups frequently split as semidirect products of smaller groups, we prove homological stability results for unitary groups over division rings with infinite centre as well as for special linear and special orthogonal groups over infinite fields.     

Character formulae for classical groups

We give formulae relating the value X_λ (g) of an irreducible character of a classical group G to entries of powers of the matrix g ε G. This yields a far-reaching generalization of a result of J.L. Cisneros-Molina concerning the GL ₂ case [1]. Partially supported by OTKA grants T 042769 and T 046365     

Minimal non-nilpotent groups as automorphism groups

Groups are classified whose automorphism group is minimal non-nilpotent.The first author wishes to thank the Mathematics Department of Napoli for its warm hospitality for the time of writing this paper.     

Finite groups of uniform logarithmic diameter

We demonstrate the existence of an infinite family of finite groups with 2 generators and logarithmic diameter with respect to any set of generators. This answers a question of A. Lubotzky. Moreover, in our groups, all minimal sets of generators have at most 3 elements. Research partially supported by NSF Grant DMS 0401006     

Statistical mechanics of classical particles with logarithmic interactions

The inhomogeneous mean-field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions. The results apply to various physical models of translation invariant plasmas, gravitating systems, as well as to planar fluid vortex motion. For attractive interactions a critical behavior occurs which can be classified as an extreme case of a second-order phase transition. To include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic-geometric mean inequality. The method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions. In addition, it also provides us with a new proof of the Trudinger-Moser inequality known from differential geometry – in its sharp form.     

Semisimple types for p-adic classical groups

We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell–Kutzko’s theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig. In an appendix, we make a correction to the proof of a result of the second author: that every irreducible cuspidal representation of a classical group as considered here is irreducibly compactly-induced from a type.     

Minimal discriminants for fields with small Frobenius groups as Galois groups

We apply class field theory to the computation of the minimal discriminants for certain solvable groups. In particular, we apply our techniques to small Frobenius groups and all imprimitive degree 8 groups such that the corresponding fields have only a degree 2 and no degree 4 subfield.     

L-functions and theta correspondence for classical groups

Using the doubling method of Piatetski-Shapiro and Rallis, we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the L-functions and local theta correspondence.     

Double Schubert polynomials for the classical groups

For each infinite series of the classical Lie groups of type B, C or D, we construct a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's Q- or P-functions defined earlier by Ivanov.