Crystals and monodromy of Bethe vectors

Halacheva, Iva; Kamnitzer, Joel; Rybnikov, Leonid; Weekes, Alex

doi:10.1215/00127094-2020-0003

Cited by 20 publications

(57 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 3.13. The corollary is a special case of a more general result of Kashiwara [10, Theorem 7.2.2], which implies that the internal cactus group action on any normal g-crystal factors over the reduced cactus group of type g (see also [7,Remark 5.21]).…”

Section: Proposition 33 [1]mentioning

confidence: 87%

“…Note that there is also a slightly different presentation of the cactus group that's often used, where generators are indexed by subdiagrams of the Dynkin diagram of a semisimple Lie algebra g [7]. The cactus group defined above corresponds to type A n−1 .…”

Section: The Cactus Group Given An Intervalmentioning

confidence: 99%

“…Proposition 2.7. [7] The cactus group C n acts on the set SSYT(λ, n) via partial Schützenberger involutions, and SYT(n, λ) is an invariant subset.…”

Section: Jmentioning

confidence: 99%

“…Indeed just as the n-strand braid group acts on n-fold tensor products of representations of U q (g) (resulting in a braided category), the cactus group C n acts on n-fold tensor products of crystals (resulting in a coboundary category) [8]. And just as the type g braid group acts on any integrable representation of U q (g), the type g cactus group acts on any g-crystal [7]. This latter "internal" action is our focus.…”

mentioning

confidence: 99%

“…In a different direction, Halacheva, Kamnitzer, Rybnikov and Weekes study the action of Gaudin algebras on tensor product multiplicity spaces [7]. Their main tool is a crystal structure on eigenvectors for shift of argument subalgebras, which are a family of commutative algebras acting on irreducible g representations.…”

mentioning

confidence: 99%

See 4 more Smart Citations

On Schützenberger modules of the cactus group

Lim¹,

Yacobi²

2021

Preprint

View full text Add to dashboard Cite

The cactus group acts on the set of standard Young tableau of a given shape by (partial) Schützenberger involutions. It is natural to extend this action to the corresponding Specht module by identifying standard Young tableau with the Kazhdan-Lusztig basis. We term these representations of the cactus group "Schützenberger modules", denoted S λ Sch , and in this paper we investigate their decomposition into irreducible components. We prove that when λ is a hook shape, the cactus group action on S λ Sch factors through Sn−1 and the resulting multiplicities are given by Kostka coefficients. Our proof relies on results of Berenstein and Kirillov and Chmutov, Glick, and Pylyavskyy.

show abstract

Section: Proposition 33 [1]mentioning

confidence: 87%

Section: The Cactus Group Given An Intervalmentioning

confidence: 99%

“…Proposition 2.7. [7] The cactus group C n acts on the set SSYT(λ, n) via partial Schützenberger involutions, and SYT(n, λ) is an invariant subset.…”

Section: Jmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

On Schützenberger modules of the cactus group

Lim¹,

Yacobi²

2021

Preprint

View full text Add to dashboard Cite

show abstract

On Some Developments of the Stokes Phenomenon

Tang

2023

Trends in Mathematics

View full text Add to dashboard Cite

Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of $${\mathcal {S}}({\mathfrak g})$$

Panyushev

Yakimova

2023

Math. Z.

View full text Add to dashboard Cite

Let $${\mathfrak g}$$ g be a semisimple Lie algebra, $$\vartheta \in \textsf{Aut}({\mathfrak g})$$ ϑ ∈ Aut ( g ) a finite order automorphism, and $${\mathfrak g}_0$$ g 0 the subalgebra of fixed points of $$\vartheta $$ ϑ . Recently, we noticed that using $$\vartheta $$ ϑ one can construct a pencil of compatible Poisson brackets on $${\mathcal {S}}({\mathfrak g})$$ S ( g ) , and thereby a ‘large’ Poisson-commutative subalgebra $${\mathcal {Z}}({\mathfrak g},\vartheta )$$ Z ( g , ϑ ) of $${\mathcal {S}}({\mathfrak g})^{{\mathfrak g}_0}$$ S ( g ) g 0 . In this article, we study invariant-theoretic properties of $$({\mathfrak g},\vartheta )$$ ( g , ϑ ) that ensure good properties of $${\mathcal {Z}}({\mathfrak g},\vartheta )$$ Z ( g , ϑ ) . Associated with $$\vartheta $$ ϑ one has a natural Lie algebra contraction $${\mathfrak g}_{(0)}$$ g ( 0 ) of $${\mathfrak g}$$ g and the notion of a good generating system (=g.g.s.) in $${\mathcal {S}}({\mathfrak g})^{\mathfrak g}$$ S ( g ) g . We prove that in many cases the equality $$\mathrm{ind\,}{\mathfrak g}_{(0)}=\mathrm{ind\,}{\mathfrak g}$$ ind g ( 0 ) = ind g holds and $${\mathcal {S}}({\mathfrak g})^{\mathfrak g}$$ S ( g ) g has a g.g.s. According to V. G. Kac’s classification of finite order automorphisms (1969), $$\vartheta $$ ϑ can be represented by a Kac diagram, $$\mathcal {K}(\vartheta )$$ K ( ϑ ) , and our results often use this presentation. The most surprising observation is that $${\mathfrak g}_{(0)}$$ g ( 0 ) depends only on the set of nodes in $$\mathcal {K}(\vartheta )$$ K ( ϑ ) with nonzero labels, and that if $$\vartheta $$ ϑ is inner and a certain label is nonzero, then $${\mathfrak g}_{(0)}$$ g ( 0 ) is isomorphic to a parabolic contraction of $${\mathfrak g}$$ g .

show abstract

Crystals and monodromy of Bethe vectors

Cited by 20 publications

References 35 publications

On Schützenberger modules of the cactus group

On Schützenberger modules of the cactus group

On Some Developments of the Stokes Phenomenon

Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of $${\mathcal {S}}({\mathfrak g})$$

Contact Info

Product

Resources

About