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Representation Theory

New submissions

[ total of 17 entries: 1-17 ]

New submissions for Thu, 28 May 20

[1]  arXiv:2005.12988 [pdf, ps, other]
Title: Algebraic Methods for Tensor Data
Subjects: Representation Theory (math.RT); Combinatorics (math.CO); Numerical Analysis (math.NA)

We develop algebraic methods for computations with tensor data. We give 3 applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and amplification of low rank tensor structure. We introduce colored Brauer diagrams, which are used for algebraic computations and in analyzing their computational complexity. We present numerical experiments whose results show that the performance of the alternating least square algorithm for the low rank approximation of tensors can be improved using tensor amplification.

[2]  arXiv:2005.13056 [pdf, other]
Title: A note on Integral Satake isomorphisms
Authors: Xinwen Zhu
Subjects: Representation Theory (math.RT)

We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an irreducible algebraic representation of $G$.

[3]  arXiv:2005.13152 [pdf, ps, other]
Title: On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate double Hecke algebras
Comments: 37 pages; Comments are welcome
Subjects: Representation Theory (math.RT)

We study the enhanced algebraic group $\uG$ of $G=\GL(V)$ over $\bbc$, which is a product variety $\GL(V)\times V$, endowed with an enhanced cross product. Associated with a natural tensor representation of $\uG$, there is naturally an enhanced Schur algebra $\cale$. We precisely investigate its structure, and study the dualities on the enhanced tensor representations for variant groups and algebras. In this course, an algebraic model of so-called degenerate double Hecke algebras (DDHA) is produced, and becomes a powerful implement. The applications of $\cale$ and DDHA give rise to two results for the classical representations of $\GL(V)$: (i) A duality between $\GL(V)$ and DDHA; (ii) A branching duality formula. With aid of the above, we further obtain an enhanced Schur-Weyl duality for $\uG$. What is more, the enhanced Schur algebra $\cale$ turns out to have only one block. The Cartan invariants for $\cale$ are precisely determined.

[4]  arXiv:2005.13172 [pdf, ps, other]
Title: Blocks with small-dimensional basic algebra
Authors: Benjamin Sambale
Comments: 13 pages

Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most 12. We extend their classification to dimension 13 and 14. As predicted by Donovan's Conjecture, we obtain only finitely many such Morita equivalence classes.

[5]  arXiv:2005.13444 [pdf, ps, other]
Title: A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
Comments: 24 pages

Building on classical invariant theory, it is observed that the polarised traces generate the centraliser $Z_L(sl(N))$ of the diagonal embedding of $U(sl(N))$ in $U(sl(N))^{\otimes L}$. The paper then focuses on $sl(3)$ and the case $L=2$. A Calabi--Yau algebra $\mathcal{A}$ with three generators is introduced and explicitly shown to possess a PBW basis and a certain central element. It is seen that $Z_2(sl(3))$ is isomorphic to a quotient of the algebra $\mathcal{A}$ by a single explicit relation fixing the value of the central element. Upon concentrating on three highest weight representations occurring in the Clebsch--Gordan series of $U(sl(3))$, a specialisation of $\mathcal{A}$ arises, involving the pairs of numbers characterising the three highest weights. In this realisation in $U(sl(3))\otimes U(sl(3))$, the coefficients in the defining relations and the value of the central element have degrees that correspond to the fundamental degrees of the Weyl group of type $E_6$. With the correct association between the six parameters of the representations and some roots of $E_6$, the symmetry under the full Weyl group of type $E_6$ is made manifest. The coefficients of the relations and the value of the central element in the realisation in $U(sl(3))\otimes U(sl(3))$ are expressed in terms of the fundamental invariant polynomials associated to $E_6$. It is also shown that the relations of the algebra $\mathcal{A}$ can be realised with Heun type operators in the Racah or Hahn algebra.

Cross-lists for Thu, 28 May 20

[6]  arXiv:2005.12954 (cross-list from math.RA) [pdf, ps, other]
Title: $V$-universal Hopf algebras (co)acting on $Ω$-algebras
Comments: 31 pages
Subjects: Rings and Algebras (math.RA); Category Theory (math.CT); Quantum Algebra (math.QA); Representation Theory (math.RT)

We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced \cite{AGV1} bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of $A$, called $\Omega$-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study $V$-universal measuring coalgebras and $V$-universal comeasuring algebras between $\Omega$-algebras $A$ and $B$, relative to a fixed subspace $V$ of $\Vect(A,B)$. By considering the case $A=B$, we derive the notion of a $V$-universal (co)acting bialgebra (and Hopf algebra) for a given algebra $A$. In particular, this leads to a refinement of the existence conditions for the Manin-Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the $V$-universal acting bi/Hopf algebra and the finite dual of the $V$-universal coacting bi/Hopf algebra under certain conditions on $V$ in terms of the finite topology on $\End_F(A)$.

[7]  arXiv:2005.12980 (cross-list from math.AG) [pdf, other]
Title: Capped vertex with descendants for zero dimensional $A_{\infty}$ quiver varieties

In this paper, we study the capped vertex functions associated to certain zero-dimensional type-$A$ Nakajima quiver varieties. The insertion of descendants into the vertex functions can be expressed by the Macdonald operators, which leads to explicit combinatorial formulas for the capped vertex functions.
We determine the monodromy of the vertex functions and show that it coincides with the elliptic R-matrix of symplectic dual variety. We apply our results to give the vertex functions and the characters of the tautological bundles on the quiver varieties formed from arbitrary stability conditions.

[8]  arXiv:2005.13381 (cross-list from math.CT) [pdf, ps, other]
Title: Classifying substructures of extriangulated categories via Serre subcategories
Authors: Haruhisa Enomoto
Comments: 12 pages, comments welcome

We give a classification of substructures (= closed subbifunctors) of a given skeletally small extriangulated category by using the category of defects, in a similar way to the author's classification of exact structures of a given additive category. More precisely, for an extriangulated category, possible substructures are in bijection with Serre subcategories of an abelian category consisting of defects of conflations. As a byproduct, we prove that for a given skeletally small additive category, the poset of exact structures on it is isomorphic to the poset of Serre subcategories of some abelian category.

[9]  arXiv:2005.13394 (cross-list from hep-th) [pdf, other]
Title: Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal{N}$=2 theories
Authors: Piotr Kucharski
Comments: 25 pages, 2 figures

We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as $F_K$ or $\hat{Z}$). Apart from assigning quivers to complements of $T^{(2,2p+1)}$ torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d $\mathcal{N}=2$ theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a $t$-deformation of all objects mentioned above.

[10]  arXiv:2005.13437 (cross-list from math.PR) [pdf, ps, other]
Title: Limit Profiles for Markov Chains
Comments: 23 pages
Subjects: Probability (math.PR); Combinatorics (math.CO); Group Theory (math.GR); Representation Theory (math.RT)

In a recent breakthrough, Teyssier [Tey20] introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the $k$-cycle shuffle, improving results of Hough [Hou16] and Berestycki, Schramm and Zeitouni [BSZ11]; the Ehrenfest urn diffusion with many urns, improving results of Ceccherini-Silberstein, Scarabotti and Tolli [CST07]; a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, improving results of Diaconis, Khare and Saloff-Coste [DKS08].

Replacements for Thu, 28 May 20

[11]  arXiv:1812.08160 (replaced) [pdf, other]
Title: Is there an analytic theory of automorphic functions for complex algebraic curves?
Authors: Edward Frenkel
Comments: 31 pages
Journal-ref: SIGMA 16 (2020), 042
Subjects: Representation Theory (math.RT); High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG); Quantum Algebra (math.QA)
[12]  arXiv:1903.12059 (replaced) [pdf, ps, other]
Title: Computation of cohomology of Lie conformal and Poisson vertex algebras
Comments: 42 pages, revised version with some notational changes
[13]  arXiv:1904.11903 (replaced) [pdf, ps, other]
Title: Stratifying systems through $τ$-tilting theory
Comments: Major copyediting of the text. No substantial mathematical changes. Accepted for publication in Documenta Mathematica
[14]  arXiv:2001.11195 (replaced) [pdf, ps, other]
Title: The Racah algebra: An overview and recent results
Comments: 18 pages, survey paper based on talk at the conference Representation Theory XVI in Dubrovnik, 2019. Version 2: some typos corrected and references updated. Accepted for publication in Contemporary Mathematics
[15]  arXiv:2003.10499 (replaced) [pdf, ps, other]
Title: New incompressible symmetric tensor categories in positive characteristic
Comments: 65 pages, latex; in v.2 new subsections 3.6, 3.7, 3.8, 4.14, 4.19, small error fixed in Corollary 4.7
Subjects: Representation Theory (math.RT); Category Theory (math.CT); Quantum Algebra (math.QA)
[16]  arXiv:1809.10148 (replaced) [pdf, other]
Title: 3d Modularity
Comments: 119 pages, 10 figures and 20 tables
Subjects: High Energy Physics - Theory (hep-th); Geometric Topology (math.GT); Number Theory (math.NT); Quantum Algebra (math.QA); Representation Theory (math.RT)
[17]  arXiv:1909.10799 (replaced) [pdf, ps, other]
Title: Crossed modular categories and the Verlinde formula for twisted conformal blocks
Comments: Major Revision: reordered Sections, streamlined the main results and the proofs, added examples and expanded expositions, comments are welcome, 96 pages
[ total of 17 entries: 1-17 ]

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