Quantum algebra characters and cluster algebras
Topics in Combinatorial Representation Theory
March 21, 2008 11:00 AM to 12:00 PM
Speakers:
Kedem, Rinat
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Abstract: |
The characters of finite-dimensional representations
of quantum affine algebras (Kirillov-Reshetikhin modules)
satisfy a relation known as the $Q$-system, a discrete
integrable equation. This results in an explicit expression for
the decomposition coefficients of tensor products of these
modules.
There is a closely related fermionic formula, posessing a
positivity property, conjectured by Kirillov and Reshetikhin
based on the Bethe ansatz solution of the Heisenberg spin
chain. Special cases of this conjecture have been proven where
there is a crystal basis.
Recently (with Di Francesco) we have proven the equality of the
two multiplicity formulas. We showed that the identity can be
rephrased as the statement that KR characters are polynomials
in the initial value variables (fundamental characters).
Recently we showed that each $Q$-system can be formulated
inside a cluster algebra. As a consequence of the Laurent
phenomenon, the full cluster algebra, with the boundary
conditions corresponding to the Kirillov-Reshetikhin system,
the cluster variables exhibit a "strong Laurent phenomenon",
and are in fact polynomials in the initial variables,
generalizing the representation-theoretical result. |
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