# Lattice Paths, Young Tableaux, and Weight Multiplicities

Lattice Paths, Young Tableaux, and Weight Multiplicities For $${\ell \geq 1}$$ ℓ ≥ 1 and $${k \geq 2}$$ k ≥ 2 , we consider certain admissible sequences of k−1 lattice paths in a colored $${\ell \times \ell}$$ ℓ × ℓ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of shape $${\lambda \vdash \ell}$$ λ ⊢ ℓ with $${l(\lambda) \leq k}$$ l ( λ ) ≤ k , which is also the number of (k + 1)k··· 21-avoiding permutations in $${S_\ell}$$ S ℓ . Finally, we apply this result to the representation theory of the affine Lie algebra $${\widehat{sl}(n)}$$ sl ^ ( n ) and show that this gives the multiplicity of certain maximal dominant weights in the irreducible highest weight $${\widehat{sl}(n)}$$ sl ^ ( n ) -module $${V(k \Lambda_0)}$$ V ( k Λ 0 ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Combinatorics Springer Journals

# Lattice Paths, Young Tableaux, and Weight Multiplicities

, Volume 22 (1) – Feb 2, 2018
10 pages

/lp/springer_journal/lattice-paths-young-tableaux-and-weight-multiplicities-BriRjpZUqq
Publisher
Springer International Publishing
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Combinatorics
ISSN
0218-0006
eISSN
0219-3094
D.O.I.
10.1007/s00026-018-0374-4
Publisher site
See Article on Publisher Site

### Abstract

For $${\ell \geq 1}$$ ℓ ≥ 1 and $${k \geq 2}$$ k ≥ 2 , we consider certain admissible sequences of k−1 lattice paths in a colored $${\ell \times \ell}$$ ℓ × ℓ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of shape $${\lambda \vdash \ell}$$ λ ⊢ ℓ with $${l(\lambda) \leq k}$$ l ( λ ) ≤ k , which is also the number of (k + 1)k··· 21-avoiding permutations in $${S_\ell}$$ S ℓ . Finally, we apply this result to the representation theory of the affine Lie algebra $${\widehat{sl}(n)}$$ sl ^ ( n ) and show that this gives the multiplicity of certain maximal dominant weights in the irreducible highest weight $${\widehat{sl}(n)}$$ sl ^ ( n ) -module $${V(k \Lambda_0)}$$ V ( k Λ 0 ) .

### Journal

Annals of CombinatoricsSpringer Journals

Published: Feb 2, 2018

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