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 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 Stability Conditions for Affine Type A

2019 ◽  
Vol 23 (5) ◽  
pp. 2079-2111 ◽  
Author(s):  
P. J. Apruzzese ◽  
Kiyoshi Igusa
Keyword(s):  
Representation Theory ◽  
Linear Stability ◽  
Stability Condition ◽  
Central Charge ◽  
Type A ◽  
Maximal Length ◽  
Stability Conditions ◽  
Background Material ◽  
Affine Type

Abstract We construct maximal green sequences of maximal length for any affine quiver of type A. We determine which sets of modules (equivalently c-vectors) can occur in such sequences and, among these, which are given by a linear stability condition (also called a central charge). There is always at least one such maximal set which is linear. The proofs use representation theory and three kinds of diagrams shown in Fig. 1. Background material is reviewed with details presented in two separate papers Igusa (2017a, b).

scholarly journals Uglov Bipartitions and Extended Young Diagrams

10.37236/8559 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nicolas Jacon
Keyword(s):  
Hecke Algebras ◽  
Type A ◽  
Young Diagrams ◽  
Canonical Bases ◽  
Arbitrary Characteristic ◽  
Affine Type

We study the class of Uglov bipartitions and prove a generalization of a conjecture by Dipper, James and Murphy. We give two consequences concerning the computation of canonical bases in affine type $A$ and the description of decomposition matrices for Hecke algebras of type $B_n$ in arbitrary characteristic.

scholarly journals PBW-type filtration on quantum groups of type A

2016 ◽  
Vol 449 ◽  
pp. 321-345 ◽  
Author(s):  
Xin Fang ◽  
Ghislain Fourier ◽  
Markus Reineke
Keyword(s):  
Quantum Groups ◽  
Type A
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