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We consider Z 2 -contractions of classical Lie algebras and the behaviour of the symmetric invariants under these contractions. It is demonstrated on three different examples how the theory of symmetric polynomials works in this invariant-theoretic problem.

We prove that there exists a unique irreducible Brauer character of height zero in any 2-blocks of the symmetric group. This generalizes the theorem of P. Fong and G.D. James that the dimension of every non-trivial 2-modular simple module of the symmetric group is even.

A module of a finite group over a finite field with a symmetric non-degenerate bilinear form which is invariant by the group action is called a symmetric module. In this paper, a characterization of indecomposable orthogonal decompositions of symmetric semisimple modules and a criterion for the hyperbolic symmetric modules are obtained, and some applications to the self-dual permutation codes are shown.

The paper deals with the z-measures on partitions with the deformation (Jack) parameters 2 or 1/2. We provide a detailed explanation of the representation-theoretic origin of these measures, and of their role in the harmonic analysis on the infinite symmetric group.

We prove that Martin's conjecture holds for weight 3 blocks of symmetric group algebras—that if these blocks have Abelian defect group, then their projective (indecomposable) modules have a common radical length 7.

Independently Siemons and Wagner [I.J. Siemons, O. Wagner, private communication, 1986] and Stanley [R.P. Stanley, Positivity problems and conjectures in algebraic combinatorics, in: Mathematics: Frontiers and Perspectives, Amer. Math. Soc., 2000, pp. 295–319] conjecture that if a partition λ dominates its conjugate partition λ ′ then an FS n -homomorphism ψ λ : 1 N S n ( S λ ) S n ↪ 1 N S n ( S λ ′ ) S n is injective. We demonstrate a method of adding columns to the partition λ that preserves the injectivity of ψ λ .

Stanley introduced expressions for the normalized characters of the symmetric group and stated some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanley's positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results by Goulden and the present author. We shall see that the generating series $C(t)$ introduced by them is critical to our discussion., Comment: 20 pages, 2 figures, v2, minor revisions, fixed typos etc

In this paper we prove the Dipper–James conjecture that the centre of the Iwahori–Hecke algebra of type A is the set of symmetric polynomials in the Jucys–Murphy operators.

We relate with any symmetric function f(x,y)∈ C 〚x,y〛 presented as an infinite linear combination of Schur functions f(x,y)=∑m(λ1,λ2)S(λ1,λ2)(x,y) the multiplicity series M(f)=∑m(λ1,λ2)tλ1uλ2. We study the behavior of M(f) under natural combinatorial and algebraic constructions. In particular, we calculate the multiplicity series for the symmetric algebra of the irreducible GL 2 ( C ) -module corresponding to the complete symmetric function of degree 3. Our main result is that we have found the explicit form of the multiplicity series for the Hilbert (or Poincare) series of the algebra of invariants of two 3×3 matrices. As a consequence, we have precised the result of Berele on the asymptotics of the multiplicities in the trace cocharacter sequence of two 3×3 matrices.

In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula is general enough to give new expressions for all known types of Schubert polynomials. In the present paper we discuss the relationship between this formula and Stanley symmetric functions. We show that the coefficients obtained when a Stanley symmetric function is expressed in the basis of Schur functions are special cases of a class of generalized Littlewood-Richardson coefficients appearing in the formula for quiver varieties. We also prove that Fulton's universal Schubert polynomials (math.AG/9702012) satisfy the usual rule for a Schubert polynomial of a product of two permutations., 13 pages, 11 figures

In unpublished work, Macdonald gave an indirect proof that the connexion coefficients for certain symmetric functions coincide with the connexion coefficients of the class algebra of the symmetric group. We give a direct proof of this result and demonstrate the use of these functions in a number of combinatorial questions associated with ordered factorisations of permutations into factors of specified cycle-type, including factorisations considered up to commutation in the symmetric group. Several related properties of the symmetric functions are given.

The notion of weak normality we deal with in this note was first introduced by Andreotti and Bombieri [I]. Soon after Traverso introduced the notion of seminormality which coincides with the above one in the characteristic zero case. As is well known, the latter has an important role in connection with Picard groups, and is studied by many authors. Compared with the latter, we have little knowledge of the former in spite of its importance. The purpose of this note is to give a theorem which shows some utility of the notion of weak normality. Let R be a ring. A symmetric R-algebra of rank one is, by definition, an R-algebra which is isomorphic to a symmetric algebra S,(L) of some L E Pit(R) over R. Therefore an R-algebra is a symmetric R-algebra of rank one if and only if it is locally a polynomial algebra in one variable (cf. [3]). In this note we shall prove the following

Using the Kazhdan–Lusztig basis { C w ′ ( 1 ) | w ∈ S n } for the symmetric group algebra, we obtain nonnegativity properties of certain polynomials in matrix minors. In particular, we show that the application of these polynomials to Jacobi–Trudi matrices yields symmetric functions which are equal to nonnegative linear combinations of Schur functions.

Ž w This paper is motivated by the problem arising from quantum gravity 9, x. 2 of counting combinatorial types of triangulations S of a Riemann surface X with given degrees of vertices. Let us color 2-simplexes of the barycentric subdivision S9 in black and white, so that adjacent simplexes are of different color, and consider a mapping p : X a P onto the Riemann sphere P which send white simplexes onto the northern hemisphere, black simplexes onto the southern hemisphere, and centers k-simplexes of S into an equatorial point y , k s 0, 1, 2. Then deg p s 3n, n is k the number of 2-simplexes in S, and p is unramified outside y , y , y . It 0 1 2 is easy to see that points of X over y and y have ramification index 2 1 2 Ž . respectively, 3 , while rmaification indices of points over y are just 0 degrees of vertices of S. So the problem of counting triangulations reduces to the problem of counting ramified coverings p : X a Y of Riemann surfaces with given ramification indices. In Section 2 we recall a connection between ramified coverings p : X a Y of a Riemann surface Y of genus g and irreducible characters x of the Y symmetric group S , n s deg p . The starting point is the following forn w x Ž w x. mula, essentially owing to Hurwitz 6 see also 10, 4, 2