Cyclic sieving, rotation, and geometric representation theory. (English) Zbl 1295.22018
Let \(X\) be a finite set and \(c \in S_X\) be a permutation of \(X\) of order \(r\). We can study the sizes of the fixed point set \(X^{c^d}\) for \(d\geq 0\). Let \(f(q)\) be a polynomial. Following Reiner-Stanton-White, we say that \((X,c,f(q))\) exhibits the cyclic sieving phenomenon (CSP) if \(f(\zeta^d)=|X^{c^d}|\) for all \(d\geq 0\), where \(\zeta\) is a fixed primitive \(r\)th root of unity.
Rhoades proved that the set of semistandard Young tableaux of fixed rectangular shape and fixed content exhibits the CSP.
The goal of the present paper is to generalize Rhoades’ result, simplify his proof, and situate the result in the context of geometric representation theory.
Rhoades proved that the set of semistandard Young tableaux of fixed rectangular shape and fixed content exhibits the CSP.
The goal of the present paper is to generalize Rhoades’ result, simplify his proof, and situate the result in the context of geometric representation theory.
Reviewer: Heinrich Reitberger (Innsbruck)
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 22E57 | Geometric Langlands program: representation-theoretic aspects |
| 05E10 | Combinatorial aspects of representation theory |
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