29 research outputs found
An Analysis of the Multiplicity Spaces in Branching of Symplectic Groups
Branching of symplectic groups is not multiplicity-free. We describe a new
approach to resolving these multiplicities that is based on studying the
associated branching algebra . The algebra is a graded algebra whose
components encode the multiplicities of irreducible representations of
in irreducible representations of . Our first theorem
states that the map taking an element of to its principal submatrix induces an isomorphism of \B to a different branching
algebra \B'. The algebra \B' encodes multiplicities of irreducible
representations of in certain irreducible representations of
. Our second theorem is that each multiplicity space that arises in
the restriction of an irreducible representation of to is
canonically an irreducible module for the -fold product of . In
particular, this induces a canonical decomposition of the multiplicity spaces
into one dimensional spaces, thereby resolving the multiplicities.Comment: 32 pages, revised abstract and introduction, and reorganized the
structure of the pape
A Basis for the Symplectic Group Branching Algebra
The symplectic group branching algebra, B, is a graded algebra whose
components encode the multiplicities of irreducible representations of
Sp(2n-2,C) in each irreducible representation of Sp(2n,C). By describing on B
an ASL structure, we construct an explicit standard monomial basis of B
consisting of Sp(2n-2,C) highest weight vectors. Moreover, B is known to carry
a canonical action of the n-fold product SL(2) \times ... \times SL(2), and we
show that the standard monomial basis is the unique (up to scalar) weight basis
associated to this representation. Finally, using the theory of Hibi algebras
we describe a deformation of Spec(B) into an explicit toric variety.Comment: 20 pages, v
Quantum Polynomial Functors
We construct a category of quantum polynomial functors which deforms
Friedlander and Suslin's category of strict polynomial functors. The main aim
of this paper is to develop from first principles the basic structural
properties of this category (duality, projective generators, braiding etc.) in
analogy with classical strict polynomial functors. We then apply the work of
Hashimoto and Hayashi in this context to construct quantum Schur/Weyl functors,
and use this to provide new and easy derivations of quantum
duality, along with other results in quantum invariant theory.Comment: 34 pages, final version to appear in Journal of Algebr
A quantum Mirkovi\'c-Vybornov isomorphism
We present a quantization of an isomorphism of Mirkovi\'c and Vybornov which
relates the intersection of a Slodowy slice and a nilpotent orbit closure in
, to a slice between spherical Schubert varieties in the
affine Grassmannian of (with weights encoded by the Jordan types of the
nilpotent orbits). A quantization of the former variety is provided by a
parabolic W-algebra and of the latter by a truncated shifted Yangian. Building
on earlier work of Brundan and Kleshchev, we define an explicit isomorphism
between these non-commutative algebras, and show that its classical limit is a
variation of the original isomorphism of Mirkovi\'c and Vybornov. As a
corollary, we deduce that the W-algebra is free as a left (or right) module
over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and
Ovsienko.Comment: v2: 48 pages. Major rewrite following referee comments. Added proof
of a conjecture of Futorny, Molev, and Ovsienko that the finite W-algebra is
free over its Gelfand-Tsetlin subalgebr