Descent sets for symplectic groups

Rubey, Martin; Sagan, Bruce E.; Westbury, Bruce W.

doi:10.1007/s10801-013-0483-4

Cited by 3 publications

(8 citation statements)

References 14 publications

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“…There exists a similar (but less complicated) definition for descents in oscillating tableaux, which are used in the symplectic case instead of vacillating tableaux, and which Sundaram's bijection preserves. Thus there also exists a similar quasi-symmetric expansion of the Frobenius character, obtained for the symplectic group by Rubey, Sagan and Westbury in [9].…”

Section: Introductionsupporting

confidence: 72%

“…Moreover we introduce descent sets for vacillating tableau (see Section 2.3). We show that our bijection preserves these descents, and follow the approach taken by Rubey, Sagan and Westbury [9] for the symplectic group. This enables us to describe the quasi-symmetric expansion of the Frobenius character (see the textbook by Stanley [10]).…”

Section: Introductionsupporting

confidence: 67%

“…Thus there also exists a similar quasi-symmetric expansion of the Frobenius character. Rubey, Sagan and Westbury obtained this result in [9].…”

Section: Introductionmentioning

confidence: 58%

“…Moreover, we go further and follow the approach taken by Rubey, Sagan and Westbury [9] for the symplectic group, by introducing descent sets for vacillating tableaux; see Section 2.3. This enables us to describe the quasi-symmetric expansion of the Frobenius character.…”

Section: Schur-weyl Dualitymentioning

confidence: 99%

“…Following Rubey, Sagan and Westbury [9] we define the descent set for standard Young tableaux and vacillating tableaux.…”

Section: Descentsmentioning

confidence: 99%

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A Sundaram type Bijection for SO(3): Vacillating Tableaux and Pairs of Standard Young Tableaux and Orthogonal Littlewood-Richardson Tableaux

Jagenteufel

2018

Electron. J. Combin.

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We present a bijection between vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau for the special orthogonal group SO(2k + 1). This bijection is motivated by the direct-sum-decomposition of the rth tensor power of the defining representation of SO(2k + 1). To formulate it, we use Kwon's orthogonal Littlewood-Richardson tableaux and introduce new alternative tableaux they are in bijection with.Moreover we use a suitably defined descent set for vacillating tableaux to determine the quasi-symmetric expansion of the Frobenius characters of the isotypic components.

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Section: Introductionsupporting

confidence: 72%

Section: Introductionsupporting

confidence: 67%

“…Thus there also exists a similar quasi-symmetric expansion of the Frobenius character. Rubey, Sagan and Westbury obtained this result in [9].…”

Section: Introductionmentioning

confidence: 58%

Section: Schur-weyl Dualitymentioning

confidence: 99%

“…Following Rubey, Sagan and Westbury [9] we define the descent set for standard Young tableaux and vacillating tableaux.…”

Section: Descentsmentioning

confidence: 99%

See 3 more Smart Citations

A Sundaram type Bijection for SO(3): Vacillating Tableaux and Pairs of Standard Young Tableaux and Orthogonal Littlewood-Richardson Tableaux

Jagenteufel

2018

Electron. J. Combin.

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A combinatorial approach to classical representation theory

Rubey¹,

Westbury²

2014

Preprint

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A fundamental problem from invariant theory is to describe, given a representation V of a group G, the algebra End G (⊗ r V ) of multilinear functions invariant under the action of the group. According to Weyl's classic, a first main (later: 'fundamental') theorem of invariant theory provides a finite spanning set for this algebra, whereas a a second main theorem describes the linear relations between those basic invariants.Here we use diagrammatic methods to carry Weyl's programme a step further, providing explicit bases for the subspace of ⊗ r V invariant under the action of G, that are additionally preserved by the action of the long cycle of the symmetric group S r .The representations we study are essentially those that occur in Weyl's book, namely the defining representations of the symplectic groups, the defining representations of the symmetric groups considered as linear representations, and the adjoint representations of the linear groups.In particular, we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic groups Sp(2n). Our formulation is completely explicit and provides a very precise link to (n+1)-noncrossing perfect matchings, going beyond a dimension count. Extending our argument to the k-th symmetric powers of these representations, the combinatorial objects involved turn out to be (n + 1)-noncrossing k-regular graphs. As corollaries we obtain instances of the cyclic sieving phenomenon for these objects and the natural rotation action.In general, we are able to derive branching rules for the diagram algebras corresponding to the representations in a uniform way. Moreover, we compute the Frobenius characteristics of modules of the diagram algebras restricted to the action of the symmetric group. Via a general theorem, we also obtain the isotypic decomposition of ⊗ r V when n is large enough in comparison to r.

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Combinatorics of symplectic invariant tensors

Rubey¹,

Westbury²

2015

Preprint

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An important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant theory states that all invariants are expressible in terms of a finite number among them, whereas a second main theorem determines the relations between those basic invariants.Here we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic group Sp(2n). Our formulation is completely explicit and provides a very precise link to (n + 1)noncrossing perfect matchings, going beyond a dimension count. As a corollary, we obtain an instance of the cyclic sieving phenomenon. Résumé.Une probléme importante de la théorie des invariantes est de décrire le sous espace d'une puissance tensorielle d'une répresentation invariant à l'action de la groupe. Suivant la classique de Weyl, la théoreme fondamentale premiere pour la répresentation standard de la group sympléctique dit que toutes invariantes peuvent être expriment entre un nombre fini d'entre eux. Ainsi, une théoreme fondamentale seconde determine les rélations entre ces invariantes basiques.Ici, nous présentons une preuve transparente d'une théoreme fondamentale seconde pour la répresentation standard de la groupe sympléctique Sp(2n). Notre formulation est completement explicite est elle provide un lien tres précis avec les couplages parfaites (n + 1)-noncroissants, plus précis qu'un denombrement de la dimension. Comme corollaire nous exhibons une phénomène du crible cyclique.

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Descent sets for symplectic groups

Cited by 3 publications

References 14 publications

A Sundaram type Bijection for SO(3): Vacillating Tableaux and Pairs of Standard Young Tableaux and Orthogonal Littlewood-Richardson Tableaux

A Sundaram type Bijection for SO(3): Vacillating Tableaux and Pairs of Standard Young Tableaux and Orthogonal Littlewood-Richardson Tableaux

A combinatorial approach to classical representation theory

Combinatorics of symplectic invariant tensors

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