51 research outputs found
Invariant tensors and the cyclic sieving phenomenon
We construct a large class of examples of the cyclic sieving phenomenon by
expoiting the representation theory of semi-simple Lie algebras. Let be a
finite dimensional representation of a semi-simple Lie algebra and let be
the associated Kashiwara crystal. For , the triple which
exhibits the cyclic sieving phenomenon is constructed as follows: the set
is the set of isolated vertices in the crystal ; the map is a generalisation of promotion acting on standard tableaux of
rectangular shape and the polynomial is the fake degree of the Frobenius
character of a representation of related to the natural action
of on the subspace of invariant tensors in .
Taking to be the defining representation of gives the
cyclic sieving phenomenon for rectangular tableaux
Invariant tensors and cellular categories
Let U be the quantised enveloping algebra associated to a Cartan matrix of
finite type. Let W be the tensor product of a finite list of highest weight
representations of U. Then the centraliser algebra of W has a basis called the
dual canonical basis which gives an integral form. We show that this integral
form is cellular by using results due to Lusztig.Comment: 6 pages; to appear in Journal of Algebr
The equality of 3-manifold invariants
The invariants of 3-manifolds defined by Kuperberg for involutory Hopf
algebras and those defined by the authors for spherical Hopf algebras are the
same for Hopf algebras on which they are both defined.Comment: 8 pages, definition of state sum invariant improved for clarity, plus
minor typos corrected. With 3 postscript figures. further change: BoxedEPSF
macro now include
A universal plane of diagrammatic categories
We introduce quantum skein relations for a family of ribbon categories
parametrised by the projective plane over \bQ.
There are thirteen points for which the ribbon category admits a ribbon
functor to a category of invariant tensors
for a quantised enveloping algebra. These thirteen points lie on three
projective lines. One line gives the first
row of the Freudenthal magic square and another gives the fourth row which is
the exceptional series
Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
Using Henriques' and Kamnitzer's cactus groups, Sch\"utzenberger's promotion
and evacuation operators on standard Young tableaux can be generalised in a
very natural way to operators acting on highest weight words in tensor products
of crystals.
For the crystals corresponding to the vector representations of the
symplectic groups, we show that Sundaram's map to perfect matchings intertwines
promotion and rotation of the associated chord diagrams, and evacuation and
reversal. We also exhibit a map with similar features for the crystals
corresponding to the adjoint representations of the general linear groups.
We prove these results by applying van Leeuwen's generalisation of Fomin's
local rules for jeu de taquin, connected to the action of the cactus groups by
Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted
correspondence
Descent sets for symplectic groups
The descent set of an oscillating (or up-down) tableau is introduced. This
descent set plays the same role in the representation theory of the symplectic
groups as the descent set of a standard tableau plays in the representation
theory of the general linear groups. In particular, we show that the descent
set is preserved by Sundaram's correspondence. This gives a direct
combinatorial interpretation of the branching rules for the defining
representations of the symplectic groups; equivalently, for the Frobenius
character of the action of a symmetric group on an isotypic subspace in a
tensor power of the defining representation of a symplectic group.Comment: 22 pages, 2 figure
- …