On the representation theory of wreath products of finite groups and symmetric groups

Let G S_N be the wreath product of a finite group G and the symmetric group S_N. The aim of this paper is to prove the branching theorem for the increasing sequence of finite groups G S₁ G S₂ ... G S_N ... and the analog of Young's orthogonal form for this case, using the inductive approach invented by A. Vershik and A. Okounkov for the case of symmetric group.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 229–244.     

A new approach to standard monomial theory for classical groups

We describe a procedure for constructing monomial bases for finite dimensional irreducible representations of complex semisimple Lie algebras. A basis is calledmonomial if each of its elements is the result of applying to a (fixed) highest weight vector a monomial in the Chevalley basis elementsY , a simple root, in the opposite Borel subalgebra. As an immediate application we obtain a new proof of the main theorem of standard monomial theory for classical groups.     

A new approach to the representation theory of semisimple Lie algebras and quantum algebras

We propose a method for explicitly constructing the simple-root generators in an arbitrary finite-dimensional representation of a semisimple quantum algebra or Lie algebra. The method is based on general results from the global theory of representations of semisimple groups. The rank-two algebras A₂, B₂=C₂, D₂, and G₂ are considered as examples. The simple-root generators are represented as solutions of a system of finite-difference equations and are given in the form of N_l×N_l matrices, where N_l is the dimension of the representation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 264–284, May, 2000.     

A new characterization of alternating and symmetric groups

In this paper we prove that the alternating groupsA _n , forn=p, p+1, p+2 and symmetric groupsS _n , forn=p, p+1, wherep>=3 is a prime number, can be uniquely determined by their order components. As one of the important consequence of this characterization we show that the simple groupsA _n , wheren=p, p+1, p+2 andp>=3 is prime, satisfy in Thompson's conjecture and Shi's conjecture.     

Decomposition of the conjugacy representation of the symmetric groups

Consider the two natural representations of the symmetric groupS _n on the group algebra ℂ[S _n ]: the regular representation and the conjugacy representation (acting on the basis by conjugation). Letm(λ) be the multiplicity of the irreducible representationS ^λ in the conjugacy representation and letf ^λ be the multiplicity ofS ^λ in the regular representation. By the character estimates of [R1] and [Wa] we prove

The representation dimension of Hecke algebras and symmetric groups

We establish a lower bound for the representation dimension of all the classical Hecke algebras of types A, B and D. For all the type A algebras, and “most” of the algebras of types B and D, we also establish upper bounds. Moreover, we establish bounds for the representation dimension of group algebras of some symmetric groups.     

A new approach to the cascade theory

The cascade theory of cosmic rays is investigated from a new point of view; we deal with the number of particles produced in a certain thickness of matter, the energy of each particle being greater than E at the point of its production. This new approach leads to an elegant asymptotic theory for large thicknesses and it is better adapted to the interpretation of experiments in nuclear emulsions.     

A topological approach to the theory of groups acting on trees

Bass and Serre recast the foundations of combinatorial group theory in [7]. Here we apply the allied notions of fundamental group and covering space to redevelop their theory in a less combinatorial fashion; for example the Bass—Serre Structure Theorem is proved with no a priori knowledge of the group theoretic structure of the fundamental group of a graph of groups. Van Kampen's Theorem is used only once, in its simplest form (in the proof of Theorem 7). Cancellation arguments and normal form theorems, such as Britton's Lemma, are completely avoided; indeed they are incidental corollaries from our viewpoint. The tree which plays a central role in [7] appears in Theorem 2 as the natural analogue of the “strecken komplexe” introduced by A. Speiser [8], and subsequently also employed by R. Nevanlinna [5], to describe certain simply connected Riemann surfaces occurring in value distribution theory.     

A new approach to probability theory and thermodynamics

A new point of equilibrium thermodynamics is obtained as a consequence of the constancy of the gas density inside a closed vessel.     

A new approach to rank one linear algebraic groups

One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups.     

A new degree bound for vector invariants of symmetric groups

Let be a commutative ring, a finitely generated free -module and a finite group acting naturally on the graded symmetric algebra . Let denote the minimal number , such that the ring of invariants can be generated by finitely many elements of degree at most .

For and , the -fold direct sum of the natural permutation module, one knows that , provided that is invertible in . This was used by E. Noether to prove if .

In this paper we prove for arbitrary commutative rings and show equality for a prime power and or any ring with . Our results imply

for any ring with .

     

Computations in generic representation theory: maps from symmetric powers to composite functors

If is the finite field of order and characteristic , let be the category whose objects are functors from finite dimensional -vector spaces to -vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in include the families , and , with , defined by ,, , , and .

Fixing , we discuss the problem of computing , for all , given knowledge of for all . When , we get a complete answer for any functor chosen from the families listed above.

Our techniques involve Steenrod algebra technology, and, indeed, our most striking example, when , arose in recent work on the homology of iterated loopspaces.

     

A new duality theory for compact groups

Summary The Tannaka-Krein duality theory characterizes the category (G) of finite-dimensional, continuous, unitary representations of a compact group as a subcategory of the category of Hilbert spaces. We prove a more powerful result characterizing (G) as an abstract category: every strict symmetric monoidalC ^*-category with conjugates which has subobjects and direct sums and for which theC ^*-algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category (G) for a compact groupG unique up to isomorphism.Research supported by the Ministero della Pubblica Istruzione and CNR-GNAFA