The Kronecker product of two Schur functions sμ and sν , denoted by sμ ∗sν, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions μ and ν. The coefficient of sλ in this product is denoted by γ λ μν , and corresponds to the multiplicity of the irreducible character χ in χχ . We use Sergeev’s Formula for a S...

After deriving inequalities on coefficients arising in the expansion of a Schur P -function in terms of Schur functions we give criteria for when such expansions are multiplicity free. From here we study the multiplicity of an irreducible spin character of the twisted symmetric group in the product of a basic spin character with an irreducible character of the symmetric group, and determine whe...

We define a higher level version of the affine Hecke algebra and prove that, after completion, this is isomorphic to completion Webster's tensor product type A. then introduce Schur establish, again an isomorphism with quiver algebra. An important observation that surjects Dipper-James-Mathas cyclotomic q-Schur Moreover, we give nice diagrammatic presentations for all algebras introduced in pap...

The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan-Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming from the Chern character. Like the...

The Schur group of a commutative ring, R, with identity consists of all classes in the Brauer group of R which contain a homomorphlc image of a group ring RG for some finite group G. It is the purpose of ths article to continue an investigation of this group which was introduced in earler work as a natural gene=allzatlon of the Schur group of a field. We generalize certain facts pertaining to t...

If the complex Schur decomposition is used to solve a real linear system, then the computed solution generally has a complex component because of roundoff error. We show that the real part of the computed solution that is obtained in this way solves a nearby real linear system. Thus, it is “numerically safe” to obtain real solutions to real linear systems via the complex Schur decomposition. Th...

The Heisenberg product is an associative defined on symmetric functions which interpolates between the usual and Kronecker product. In 1938, Murnaghan discovered that of two Schur stabilizes. We prove analogous result for functions.

The product sμsν of two Schur functions is one of the most famous examples of a Schur-positive function, i.e. a symmetric function which, when written as a linear combination of Schur functions, has all positive coefficients. We ask when expressions of the form sλsρ − sμsν are Schur-positive. This general question seems to be a difficult one, but a conjecture of Fomin, Fulton, Li and Poon says ...