### Top 7 Arxiv Papers Today in Representation Theory

##### #1. Classical limit of quantum Borcherds-Bozec algebras
###### Zhaobing Fan, Seok-Jin Kang, Young-Rock Kim, Bolun Tong
Let $\mathfrak{g}$ be a Borcherds-Bozec algebra, $U(\mathfrak{g})$ be its universal enveloping algebra and $U_{q}(\mathfrak{g})$ be the corresponding quantum Borcherds-Bozec algebra. We show that the classical limit of $U_{q}(\mathfrak{g})$ is isomorphic to $U(\mathfrak{g})$ as Hopf algebras. Thus $U_{q}(\mathfrak{g})$ can be regarded as a quantum deformation of $U(\mathfrak{g})$. We also give explicit formulas for the commutation relations among the generators of $U_{q}(\mathfrak{g})$.
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###### Tweets mathRTb: Zhaobing Fan, Seok-Jin Kang, Young-Rock Kim, Bolun Tong : Classical limit of quantum Borcherds-Bozec algebras https://t.co/fz0qQK3rmd https://t.co/IiwW52H4H8
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##### #2. Specht module branching rules for wreath products of symmetric groups
###### Reuben Green
We review a class of modules for the wreath product S(m) wr S(n) of two symmetric groups which are analogous to the Specht modules of the symmetric group, and prove a pair of branching rules for this family of modules. These branching rules describe the behaviour of these wreath product Specht modules under restriction to the wreath products S(m-1) wr S(n) and S(m) wr S(n-1). In particular, we see that these restrictions of wreath product Specht modules have Specht module filtrations, and we obtain combinatorial interpretations of the multiplicities in these filtrations.
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###### Tweets mathRTb: Reuben Green : Specht module branching rules for wreath products of symmetric groups https://t.co/p02RGtHHuz https://t.co/0ITzTuOhwL
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##### #3. Irreducible self-adjoint representation of quantum Teichmüller space and phase constants
###### Hyun Kyu Kim
Quantization of Teichm\"uller space of a non-compact Riemann surface has emerged in 1980's as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston's shear coordinate functions on the edges form a coordinate system for the Teichm\"uller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon change of triangulations, one must construct a unitary operator between Hilbert spaces intertwining the quantum coordinate operators and satisfying composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a reducible representation, and the phase constants are all trivial. In the present paper, we employ the harmonic-analytic theory of Shale-Weil intertwiners for Schr\"odinger representations, as well as Faddeev-Kashaev's quantum dilogarithm function, to construct irreducible representations of the quantum shear coordinate functions and the...
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###### Tweets MathPHYPapers: Irreducible self-adjoint representation of quantum Teichm\"uller space and phase constants. https://t.co/hvLxtnZoes mathRTb: Hyun Kyu Kim : Irreducible self-adjoint representation of quantum Teichmüller space and phase constants https://t.co/xOtAZ3Mynn https://t.co/MkMJwPzcYb
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##### #4. Three perspectives on categorical symmetric Howe duality
###### Ben Webster
In this paper, we consider the categorical symmetric Howe duality introduced by Khovanov, Lauda, Sussan and Yonezawa. While originally defined from a purely diagrammatic perspective, this construction also has geometric and representation-theoretic interpretations, corresponding to certain perverse sheaves on spaces of quiver representations and the category of Gelfand-Tsetlin modules over $\mathfrak{gl}_n$. In particular, we show that the deformed Webster algebras'' discussed in work of Khovanov-Lauda-Sussan-Yonezawa manifest a Koszul duality between blocks of the category of Gelfand-Tsetlin modules over $\mathfrak{gl}_n$, and the constructible sheaves on representations of a linear quiver invariant under a certain parabolic in the group that acts by changing bases. Furthermore, we show that this duality intertwines translation functors with a diagrammatic categorical action. Includes an appendix by the author and Jerry Guan.
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###### Tweets mathRTb: Ben Webster : Three perspectives on categorical symmetric Howe duality https://t.co/6aPSk0GJHQ https://t.co/1mPGOsCibq
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##### #5. Simple Modules Over The Quantum Affine Space
###### Snehashis Mukherjee, Sanu Bera
The coordinate ring $\mathbb{K}_\mathcal{Q}[x_1,\cdots ,x_n]$ of the quantum affine space is the algebra generated over a field $\mathbb{K}$ by the variables $x_1,\cdots ,x_n$ satisfying the relations $x_ix_j=q_{ij}x_jx_i, \ \forall \ 1 \leq i,j\leq n$. We construct simple $\mathbb{K}_\mathcal{Q}[x_1,\cdots ,x_n]$-modules in a more general setting where the entries $q_{ij}$ lie in a torsion subgroup of $\mathbb{K}^*$ and show analogous results hold as in .
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###### Tweets mathRTb: Snehashis Mukherjee, Sanu Bera : Simple Modules Over The Quantum Affine Space https://t.co/RK00wPoDY7 https://t.co/VSKu8h02zx
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##### #6. The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$
###### Drazen Adamovic, Thomas Creutzig, Naoki Genra, Jinwei Yang
The vertex algebras $V^{(p)}$ and $R^{(p)}$ introduced in  are very interesting relatives of the famous triplet algebras of logarithmic CFT. The algebra $V^{(p)}$ (respectively, $R^{(p)}$) is a large extension of the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ (respectively, $L_k(\mathfrak{sl}_2)$ times a Heisenberg algebra), at level $k=-2+1/p$ for positive integer $p$. In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $V^{(p)}$ and we decompose $V^{(p)}$ as an $L_k(\mathfrak{sl}_2)$-module and $R^{(p)}$ as an $L_k(\mathfrak{gl}_2)$-module. The decomposition of $V^{(p)}$ shows that $V^{(p)}$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $V^{(p)}$ is the logarithmic doublet algebra $A^{(p)}$ introduced in , while the reduction of $R^{(p)}$ yields the $B^{(p)}$-algebra of...
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###### Tweets mathRTb: Drazen Adamovic, Thomas Creutzig, Naoki Genra, Jinwei Yang : The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$ https://t.co/gd4vJL4IZZ https://t.co/Olg0994IL6 TheoryPapers: The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$. https://t.co/imH7nFPfqX
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##### #7. On Schur problem and Kostka numbers
###### Robert Coquereaux, Jean-Bernard Zuber
We reconsider the two related problems: distribution of the diagonal elements of a Hermitian n x n matrix of known eigenvalues (Schur) and determination of multiplicities of weights in a given irreducible representation of SU(n) (Kostka). It is well known that the former yields a semi-classical picture of the latter. We present explicit expressions for low values of n that complement those given in the literature, recall some exact (non asymptotic) relation between the two problems, comment on the limiting procedure whereby Kostka numbers are obtained from Littlewood-Richardson coefficients, and finally extend these considerations to the case of the B2 algebra, with a few novel conjectures.
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###### Tweets mathRTb: Robert Coquereaux, Jean-Bernard Zuber : On Schur problem and Kostka numbers https://t.co/pGKCKcb7aZ https://t.co/MPZMyEsJNw MathPHYPapers: On Schur problem and Kostka numbers. https://t.co/0ME4rxjJW7
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