scholarly journals Toward Berenstein-Zelevinsky Data in Affine Type A, Part III: Proof of the Connectedness

2013 ◽  
pp. 361-402
Author(s):  
Satoshi Naito ◽  
Daisuke Sagaki ◽  
Yoshihisa Saito
Keyword(s):  
Type A ◽  
Affine Type

scholarly journals Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A

2017 ◽  
Vol 23 (4) ◽  
pp. 2553-2633 ◽  
Author(s):  
Hiraku Nakajima ◽  
Yuuya Takayama
Keyword(s):  
Gauge Theories ◽  
Type A ◽  
Affine Type

scholarly journals Two Maps on Affine Type $A$ Crystals and Hecke Algebras

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Nicolas Jacon
Keyword(s):  
Representation Theory ◽  
Quantum Groups ◽  
Fock Space ◽  
Hecke Algebras ◽  
Type A ◽  
Crystal Basis ◽  
Affine Type

We use the crystal isomorphisms of the Fock space to describe two maps on partitions and multipartitions which naturally appear in the crystal basis theory for quantum groups in affine type $A$ and in the representation theory of Hecke algebras of type $G(l,l,n)$. 

scholarly journals Matrix-Ball Construction of affine Robinson-Schensted correspondence

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Michael Chmutov ◽  
Pavlo Pylyavskyy ◽  
Elena Yudovina
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Type A ◽  
Group Symmetry ◽  
Dominant Weight ◽  
Dominance Condition ◽  
The Matrix ◽  
International Audience ◽  
Affine Type

International audience In his study of Kazhdan-Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson- Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combi- natorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Honeywill, we extend the algorithm to a bijection between extended affine symmetric group and triples (P, Q, ρ) where P and Q are tabloids and ρ is a dominant weight. The weights ρ get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.

scholarly journals Affine type A geometric crystal structure on the Grassmannian

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Gabriel Frieden
Keyword(s):  
Crystal Structure ◽  
Type A ◽  
Young Tableaux ◽  
Rectangular Shape ◽  
International Audience ◽  
Affine Type

International audience We construct a type A(1) n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard Young tableaux of rectangular shape (with n − k rows), including the affine crystal operator e 0. In particular, the promotion operation on these tableaux essentially corresponds to cyclically shifting the Plu ̈cker coordinates of the Grassmannian.

scholarly journals The ABC's of affine Grassmannians and Hall-Littlewood polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Avinash J. Dalal ◽  
Jennifer Morse
Keyword(s):  
Schur Functions ◽  
Type A ◽  
Combinatorial Formula ◽  
Transition Matrices ◽  
Nous Obtenons ◽  
Affine Grassmannians ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Schubert Classes ◽  
Affine Type

International audience We give a new description of the Pieri rule for $k$-Schur functions using the Bruhat order on the affine type-$A$ Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and $k$-Schur functions. Nous présentons une nouvelle description, issue de l'ordre de Bruhat du groupe de Weyl affine de type $A$, de la règle de Pieri pour les fonctions $k$-Schur. Ce faisant, nous obtenons une nouvelle formule combinatoire pour les représentants des classes de Schubert de la cohomologie des Grassmannienne affines. Nous décrivons aussi comment notre approche permet d'obtenir les polynômes de Kostka-Foulkes et comment elle peut être appliquée à l’étude des matrices de transition entre les polynômes de Hall-Littlewood et les fonctions $k$-Schur.

scholarly journals Fock Space Representation of the Circle Quantum Group

2019 ◽  
Author(s):  
Francesco Sala ◽  
Olivier Schiffmann
Keyword(s):  
Vector Space ◽  
Quantum Group ◽  
Lie Algebras ◽  
Quantum Groups ◽  
Fock Space ◽  
Type A ◽  
Space Representation ◽  
Affine Type

Abstract In [12] we have defined quantum groups $\mathbf{U}_{\upsilon }(\mathfrak{sl}(\mathbb{R}))$ and $\mathbf{U}_{\upsilon }(\mathfrak{sl}(S^1))$, which can be interpreted as continuum generalizations of the quantum groups of the Kac–Moody Lie algebras of finite, respectively affine type $A$. In the present paper, we define the Fock space representation $\mathcal{F}_{\mathbb{R}}$ of the quantum group $\mathbf{U}_{\upsilon }(\mathfrak{sl}(\mathbb{R}))$ as the vector space generated by real pyramids (a continuum generalization of the notion of partition). In addition, by using a variant version of the “folding procedure” of Hayashi–Misra–Miwa, we define an action of $\mathbf{U}_{\upsilon }(\mathfrak{sl}(S^1))$ on $\mathcal{F}_{\mathbb{R}}$.

scholarly journals Categorical action of the extended braid group of affine type A

2017 ◽  
Vol 19 (03) ◽  
pp. 1650024 ◽  
Author(s):  
Agnès Gadbled ◽  
Anne-Laure Thiel ◽  
Emmanuel Wagner
Keyword(s):  
Braid Group ◽  
Type A ◽  
Homotopy Category ◽  
Projective Modules ◽  
Intersection Numbers ◽  
Finitely Generated ◽  
Type Formula ◽  
Affine Type ◽  
Cyclic Quiver

Using a quiver algebra of a double cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type [Formula: see text] on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with intersection numbers coming from the topological origin of the group.

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