A web basis of invariant polynomials from noncrossing partitions. (English) Zbl 1504.05023
Specht modules \(S^\lambda\), indexed by integer partitions \(\lambda\), are irreducible complex representations of symmetric groups \(S_n\). There are many different ways to construct Specht modules. These various constructions yield distinct linear bases of \(S^\lambda\). An important construction of the Specht module might be the class of global sections of a line bundle on a partial flag variety. In this case, standard monomial theory endows each \(S^\lambda\) with a natural basis of polynomials, encoded by standard Young tableaux of shape \(\lambda\). In special cases, this realization of \(S^\lambda\) has also a remarkable basis, consisting of a different set of invariant polynomials, encoded by planar diagrams called webs.
From the authors’ abstract: “Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases – essentially, only the cases \(\lambda=(d,d)\) and \(\lambda=(d,d,d)\). Building on work of B. Rhoades [J. Algebr. Comb. 45, No. 1, 81–127 (2017; Zbl 1355.05280)], we construct an apparent web basis of invariant polynomials for the \(2\)-parameter family of Specht modules with \(\lambda\) of the form \((d,d,1^l)\). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.”
From the authors’ abstract: “Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases – essentially, only the cases \(\lambda=(d,d)\) and \(\lambda=(d,d,d)\). Building on work of B. Rhoades [J. Algebr. Comb. 45, No. 1, 81–127 (2017; Zbl 1355.05280)], we construct an apparent web basis of invariant polynomials for the \(2\)-parameter family of Specht modules with \(\lambda\) of the form \((d,d,1^l)\). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.”
Reviewer: Ljuben Mutafchiev (Sofia)
MSC:
| 05E10 | Combinatorial aspects of representation theory |
| 05A18 | Partitions of sets |
| 20C30 | Representations of finite symmetric groups |
Keywords:
Specht module; set partition; invariant polynomial; web basis; standard Young tableau; increasing tableau; cyclic sieving phenomenonCitations:
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