194 research outputs found
Identities from representation theory
We give a new Jacobi--Trudi-type formula for characters of finite-dimensional
irreducible representations in type using characters of the fundamental
representations and non-intersecting lattice paths. We give equivalent
determinant formulas for the decomposition multiplicities for tensor powers of
the spin representation in type and the exterior representation in type
. This gives a combinatorial proof of an identity of Katz and equates such
a multiplicity with the dimension of an irreducible representation in type
. By taking certain specializations, we obtain identities for -Catalan
triangle numbers, the -Catalan number of Stump, -triangle versions of
Motzkin and Riordan numbers, and generalizations of Touchard's identity. We use
(spin) rigid tableaux and crystal base theory to show some formulas relating
Catalan, Motzkin, and Riordan triangle numbers.Comment: 68 pages, 8 figure
Riordan Paths and Derangements
Riordan paths are Motzkin paths without horizontal steps on the x-axis. We
establish a correspondence between Riordan paths and
-avoiding derangements. We also present a combinatorial proof
of a recurrence relation for the Riordan numbers in the spirit of the
Foata-Zeilberger proof of a recurrence relation on the Schr\"oder numbers.Comment: 9 pages, 2 figure
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