Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.

We find that a compatible graded left-symmetric algebra structure on the Witt algebra induces an indecomposable module of the Witt algebra with 1-dimensional weight spaces by its left multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebra structures on the Witt algebra are classified. All of them are simple and they in...

Let M”= F-'(m,) be the preimage of the monomial symmetric function m, under F. We give an interpretation of M” in terms of certain combinatorial structures called I-domino tabloids. Using this interpretation, a number of properties of M” can be derived. The combinatorial interpretation of the preimage of the so called forgotten basis of Doubilet and Rota can also be obtained by similar techniques.

An equivalent relation between transitive 1-factorizations of arctransitive graphs and factorizations of their automorphism groups is established. This relation provides a method for constructing and characterizing transitive 1-factorizations for certain classes of arc-transitive graphs. Then a characterization is given of 2-arc-transitive graphs which have transitive 1factorizations. In this c...

We prove generalized arithmetic-geometric mean inequalities for quasi-means arising from symmetric polynomials. The inequalities are satisfied by all positive, homogeneous symmetric polynomials, as well as a certain family of nonhomogeneous polynomials; this family allows us to prove the following combinatorial result for marked square grids. Suppose that the cells of a n × n checkerboard are e...

In this paper we introduce a family of polynomials indexed by pairs of partitions and show that if these polynomials are self–orthogonal then the centre of the Iwahori–Hecke algebra of the symmetric group is precisely the set of symmetric polynomials in the Murphy operators.

Abstract. This paper will introduce noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand, one can derive a noncommutative Cauchy identity as well as study a noncommutative scalar product implied by Cauchy identity. This scalar product ...

We show how the superembedding formalism can be applied to construct manifestly kappasymmetric higher derivative corrections for the D9-brane. We also show that all correction terms appear at even powers of the fundamental length scale l. We explicitly construct the first potential correction, which corresponds to the kappa-symmetric version of the ∂4F 4, which one finds from the four-point amp...

An explicit expansion for the Macdonald polynomials of the form H (32 a 1 b) [X; q, t] and H (41 k) [X; q, t] in terms of Hall-Littlewood symmetric functions is presented. The expansion gives a proof of a symmetric function operator that adds a row of size 3 to Macdonald polynomials of the form H (2 a 1 b) [X; q, t] and another operator that adds a row of size 4 to Macdonald polynomials of only...