23 research outputs found
Stabilized plethysms for the classical Lie groups
The plethysms of the Weyl characters associated to a classical Lie group by
the symmetric functions stabilize in large rank. In the case of a power sum
plethysm, we prove that the coefficients of the decomposition of this
stabilized form on the basis of Weyl characters are branching coefficients
which can be determined by a simple algorithm. This generalizes in particular
some classical results by Littlewood on the power sum plethysms of Schur
functions. We also establish explicit formulas for the outer multiplicities
appearing in the decomposition of the tensor square of any irreducible finite
dimensional module into its symmetric and antisymmetric parts. These
multiplicities can notably be expressed in terms of the Littlewood-Richardson
coefficients
A duality between -multiplicities in tensor products and -multiplicities of weights for the root systems or
Starting from Jacobi-Trudi's type determinental expressions for the Schur
functions corresponding to types and we define a natural
-analogue of the multiplicity when is a
tensor product of row or column shaped modules defined by . We prove that
these -multiplicities are equal to certain Kostka-Foulkes polynomials
related to the root systems or . Finally we derive formulas expressing
the associated multiplicities in terms of Kostka numbers
Parabolic Kazhdan-Lusztig polynomials, plethysm and gereralized Hall-Littlewood functions for classical types
We use power sums plethysm operators to introduce H functions which
interpolate between the Weyl characters and the Hall-Littlewood functions Q'
corresponding to classical Lie groups. The coefficients of these functions on
the basis of Weyl characters are parabolic Kazhdan-Lusztig polynomials and
thus, are nonnegative. We prove that they can be regarded as quantizations of
branching coefficients obtained by restriction to certain Levi subgroups. The H
functions associated to linear groups coincide with the polynomials introduced
by Lascoux Leclerc and Thibon (LLT polynomials).Comment: To appear in European Journal of Combinatoric