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Identities from representation theory
- Source :
- Discrete Mathematics. 342:2493-2541
- Publication Year :
- 2019
- Publisher :
- Elsevier BV, 2019.
-
Abstract
- We give a new Jacobi--Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ 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 $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identities for $q$-Catalan triangle numbers, the $q,t$-Catalan number of Stump, $q$-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.<br />Comment: 68 pages, 8 figures
- Subjects :
- Combinatorial proof
0102 computer and information sciences
02 engineering and technology
01 natural sciences
Representation theory
Theoretical Computer Science
Catalan number
Combinatorics
FOS: Mathematics
0202 electrical engineering, electronic engineering, information engineering
Mathematics - Combinatorics
Discrete Mathematics and Combinatorics
Representation Theory (math.RT)
Mathematics
Mathematics::Combinatorics
020206 networking & telecommunications
Multiplicity (mathematics)
16. Peace & justice
Spin representation
010201 computation theory & mathematics
Irreducible representation
q-analog
05E10, 05A19
Combinatorics (math.CO)
Mathematics - Representation Theory
Crystal base
Subjects
Details
- ISSN :
- 0012365X
- Volume :
- 342
- Database :
- OpenAIRE
- Journal :
- Discrete Mathematics
- Accession number :
- edsair.doi.dedup.....cca922d1c6ca2feefeb43f8560c8207a