Some new relations on skew Schur function differences are established both combinatorially using Schützenberger’s jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain differences of skew Schur functions are Schur positive. Applying these results to a basis of symmetric functions involving ribbon Schur functions confirms the validi...

To each partition λ, we introduce a measure Sλ(x; t)sλ(y)/Zt where sλ is the Schur function and Sλ(x; t) is a generalization of the Schur function defined in [M] and Zt is a normalization constant. This measure, which we call the t-Schur measure, is a generalization of the Schur measure [O] and the shifted Schur measure studied by Tracy and Widom [TW3]. We prove that by a certain specialization...

An elimination procedure based on Schur complement calculation is proposed for restoring the nonzero elements of a sparse Jacobian matrix. We provide a sparsity usage parameter that can be specified by users to tune the cost of matrix-vector product calculation and the cost of restoration of the nonzero entries. We employ Pascal seeding to compress the Jacobian matrix. Preliminary numerical tes...

Starting from Jacobi-Trudi’s type determinental expressions for the Schur functions corresponding to types B,C and D, we define a natural q-analogue of the multiplicity [V (λ) : M(μ)] when M(μ) is a tensor product of row or column shaped modules defined by μ. We prove that these q-multiplicities are equal to certain Kostka-Foulkes polynomials related to the root systems C or D. Finally we deriv...

The Schur-Weyl Duality Theorem motivates the general deenition of dual representations. Such representations arise in a very large class of groups, including all compact, nilpotent and semidirect product groups, as well as the complementary groups in the physics literature and the reductive dual pairs in the mathematics literature. Several applications of the notion of duality are given, includ...

We give a polynomial time attack on the McEliece public key cryptosystem based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes on the distinguishability of such codes from random codes using the Schur product. Wieschebrink treated the genus zero case a few years ago but his approach cannot be extent straightforwardly to other genera. We address this problem by introduc...

One purpose is to isolate techniques and results in representation theory which do not depend upon additional structure of the groups. Much can be done in the representation theory of compact groups without anything more than the compactness. Similarly, the discrete decomposition of L(Γ\G) for compact quotients Γ\G depends upon nothing more than compactness. Schur orthogonality and inner produc...

The distributions of the smallest and largest eigenvalues for the matrix product Z † Z, where Z is an n × m complex Gaussian matrix with correlations both along rows and down columns, are expressed as m × m determinants. In the case of correlation along rows, these expressions are computationally more efficient than those involving sums over partitions and Schur polynomials reported recently fo...

We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n, K), where K is an infinite field of characteristic not 2. The bideterminants are indexed by pairs of Young tableaux which are O(n)-standard in the sense of King-Welsh. We also give an explicit filtration of K[O(n)] as an O(n, K)-bimodule, whose factors are isormorphic to the tensor product of orthogon...