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 describe a Hopf algebraic approach to the Grothendieck ring of representations of subgroups Hπ of the general linear group GL(n) which stabilize a tensor of Young symmetry {π}. It turns out that the representation ring of the subgroup can be described as a Hopf algebra twist, with a 2-cocycle derived from the Cauchy kernel 2-cocycle using plethysms. Due to Schur-Weyl duality we also need to ...

We define spatial CPD-semigroup and construct their Powers sum. We construct the Powers sum for general spatial CP-semigroups. In both cases, we show that the product system of that Powers sum is the product of the spatial product systems of its factors. We show that on the domain of intersection, pointwise bounded CPD-semigroups on the one side and Schur CP-semigroups on the other, the constru...

In this note we describe a seemingly new approach to the complex representation theory of the wreath product G o Sd where G is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphi...

Let V ⊗n be the n–fold tensor product of a vector space V. Following I. Schur we consider the action of the symmetric group Sn on V ⊗n by permuting coordinates. In the ‘super’ (Z2 graded) case V = V0 ⊕ V1, a ± sign is added [BR]. These actions give rise to the corresponding Schur algebras S(Sn, V ). Here S(Sn, V ) is compared with S(An, V ), the Schur algebra corresponding to the alternating su...

Recently, residue and quotient tables were defined by Fishel and the author, and were used to describe strong covers in the lattice of k-bounded partitions. In this paper, we show or conjecture that residue and quotient tables can be used to describe many other results in the theory of k-bounded partitions and k-Schur functions, including k-conjugates, weak horizontal and vertical strips, and t...

The classical Pieri formula is an explicit rule for determining the coefficients in the expansion s1m · sλ = ∑ c 1,λ sμ , where sν is the Schur polynomial indexed by the partition ν. Since the Schur polynomials represent Schubert classes in the cohomology of the complex Grassmannian, this gives a partial description of the cup product in this cohomology. Pieri’s formula was generalized to the c...

In [4], Gurevich, Pyatov and Saponov stated an expansion for the product of two Schur functions and gave a proof based on the Plücker relations. Here we show that this identity is in fact a special case of a quite general Schur function identity, which was stated and proved in [1, Lemma 16]. In [1], it was used to prove bijectively Dodgson’s condensation formula and the Plücker relations, but w...

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SLn and a similar formula is conjectured for S...