# Representation Theory

## New submissions

[ total of 21 entries: 1-21 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Potentials for Moduli Spaces of A_m-local Systems on Surfaces
Authors: Efim Abrikosov
Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG)

We study properties of potentials on quivers $Q_{\mathcal{T},m}$ arising from cluster coordinates on moduli spaces of $PGL_{m+1}$-local systems on a topological surface with punctures. To every quiver with potential one can associate a $3d$ Calabi-Yau $A_\infty$-category in such a way that a natural notion of equivalence for quivers with potentials (called "right-equivalence") translates to $A_\infty$-equivalence of associated categories.
For any quiver one can define a notion of a "primitive" potential. Our first result is the description of the space of equivalence classes of primitive potentials on quivers $Q_{\mathcal{T}, m}$. Then we provide a full description of the space of equivalence classes of all \emph{generic} potentials for the case $m = 2$ (corresponds to $PGL_3$-local systems). In particular, we show that it is finite-dimensional. This claim extends results of Gei\ss, Labardini-Fragoso and Schr\"oer who have proved analogous statement in $m=1$ case.
In many cases $3d$ Calabi-Yau $A_\infty$-categories constructed from quivers with potentials are expected to be realized geometrically as Fukaya categories of certain Calabi-Yau $3$-folds. Bridgeland and Smith gave an explicit construction of Fukaya categories for quivers $Q_{\mathcal{T},m=1}$. We propose a candidate for Calabi-Yau $3$-folds that would play analogous role in higher rank cases, $m > 1$. We study their (co)homology and describe a construction of collections of $3$-dimensional spheres that should play a role of generating collections of Lagrangian spheres in corresponding Fukaya categories.

[2]
Title: Cell Decompositions for Rank Two Quiver Grassmannians
Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG); Rings and Algebras (math.RA)

We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The natural bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians.

[3]
Title: Layer structure of irreducible Lie algebra modules
Authors: Jorgen Rasmussen
Subjects: Representation Theory (math.RT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Combinatorics (math.CO)

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of every finite-dimensional irreducible $\mathfrak{g}$-module admits a decomposition in terms of layer sums, with only non-negative integer coefficients. Ensuing results include a new approach to the computation of Weyl characters and weight multiplicities, and a closed-form expression for the number of distinct weights in a finite-dimensional irreducible $\mathfrak{g}$-module. The latter is given by a polynomial in the Dynkin labels, of degree equal to the rank of $\mathfrak{g}$.

[4]
Title: How to sheafify an elliptic quantum group
Comments: These lecture notes are based on Yang's talk at the MATRIX program Geometric R-Matrices: from Geometry to Probability, at the University of Melbourne, Dec.18-22, 2017, and Zhao's talk at Perimeter Institute for Theoretical Physics in January 2018. 12 pages, expository paper, submitted to the MATRIX Annals
Subjects: Representation Theory (math.RT)

We give an introductory survey of the results in arXiv: 1708.01418. We discuss a sheafified elliptic quantum group associated to any symmetric Kac-Moody Lie algebra. The sheafification is obtained by applying the equivariant elliptic cohomological theory to the moduli space of representations of a preprojective algebra. By construction, the elliptic quantum group naturally acts on the equivariant elliptic cohomology of Nakajima quiver varieties. As an application, we obtain a relation between the sheafified elliptic quantum group and the global affine Grassmannian over an elliptic curve.

[5]
Title: Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
Subjects: Representation Theory (math.RT); Quantum Algebra (math.QA); Rings and Algebras (math.RA)

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal subcategories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms are specialized at $t = 1$ to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002 : the multiplicities of simple modules in standard modules in the categories above for type $B_n^{(1)}$ are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.

[6]
Title: Short Proof of a Conjecture Concerning Split-By-Nilpotent Extensions
Authors: Stephen Zito
Subjects: Representation Theory (math.RT)

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E. We provide a short proof to a conjecture by Assem and Zacharia concerning properties of mod B inherited by mod C. We show if B is a tilted algebra, then C is a tilted algebra.

[7]
Title: Cyclic Sieving and Cluster Duality for Grassmannian
Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Combinatorics (math.CO)

We introduce a decorated configuration space $\mathscr{C}onf_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ canonically parametrizes a linear basis of the homogenous coordinate ring of the Grassmannian ${\rm Gr}_a(n)$. We prove that $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ is equivalent to the mirror Landau-Ginzburg model of Grassmannian considered by Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.

[8]
Title: Auslander-Reiten $(d+2)$-angles in subcategories and a $(d+2)$-angulated generalisation of a theorem by Brüning
Authors: Francesca Fedele
Subjects: Representation Theory (math.RT)

Let $\Phi$ be a finite dimensional algebra over an algebraically closed field $k$ and assume gldim$\,\Phi\leq d$, for some fixed positive integer $d$. For $d=1$, Br\"uning proved that there is a bijection between the wide subcategories of the abelian category mod$\,\Phi$ and those of the triangulated category $\mathcal{D}^b(\text{mod}\Phi)$. Moreover, for a suitable triangulated category $\mathcal{M}$, J{\o}rgensen gave a description of Auslander-Reiten triangles in the extension closed subcategories of $\mathcal{M}$.
In this paper, we generalise these results for $d$-abelian and $(d+2)$-angulated categories, where kernels and cokernels are replaced by complexes of $d+1$ objects and triangles are replaced by complexes of $d+2$ objects. The categories are obtained as follows: if $\mathcal{F}\subseteq \text{mod} \Phi$ is a $d$-cluster tilting subcategory, consider $\overline{\mathcal{F}}:=\text{add} \{\Sigma^{id}\mathcal{F}\mid i\in\mathbb{Z} \}\subseteq \mathcal{D}^b(\text{mod}\Phi)$. Then $\mathcal{F}$ is $d$-abelian and plays the role of a higher mod$\,\Phi$ having for higher derived category the $(d+2)$-angulated category $\overline{\mathcal{F}}$.

### Cross-lists for Tue, 20 Mar 18

[9]  arXiv:1803.06357 (cross-list from math.RA) [pdf, ps, other]
Title: Maximal subalgebras of the exceptional Lie algebras in low characteristic
Authors: Thomas Purslow
Subjects: Rings and Algebras (math.RA); Representation Theory (math.RT)

In this thesis we consider the maximal subalgebras of the exceptional Lie algebras in algebraically closed fields of positive characteristic. This begins with a quick recap of the article by Herpel and Stewart which considered the Cartan type maximal subalgebras in the exceptional Lie algebras for good characteristic, and then the article by Premet considering non-semisimple maximal subalgebras in good characteristic.
For $p=5$ we give an example of what appears to be a new maximal subalgebra in the exceptional Lie algebra of type $E_8$. We show that this maximal subalgebra is isomorphic to the $p$-closure of the non-restricted Witt algebra $W(1;2)$.
After this, we focus completely on characteristics $p=2$ and $p=3$ giving examples of new non-semisimple maximal subalgebras in the exceptional Lie algebras. We consider the Weisfeiler filtration associated to these maximal subalgebras and leave many open questions. There are one or two examples of simple maximal subalgebras in $F_4$ for $p=3$ and $E_8$ for $p=2$.

[10]  arXiv:1803.06463 (cross-list from math.QA) [pdf, ps, other]
Title: Multiplication formulas and semisimplicity for q-Schur superalgebras
Comments: 22 pages. Nagoya J. Math. (to appear)
Subjects: Quantum Algebra (math.QA); Group Theory (math.GR); Rings and Algebras (math.RA); Representation Theory (math.RT)

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative norm approach developed by the first two authors. We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for q-Schur superalgebras. We also construct infinitesimal and little q-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

[11]  arXiv:1803.06515 (cross-list from quant-ph) [pdf, ps, other]
Title: From nonholonomic quantum constraint to canonical variables of photons I: true intrinsic degree of freedom
Subjects: Quantum Physics (quant-ph); Representation Theory (math.RT); Optics (physics.optics)

We report that the true intrinsic degree of freedom of the photon is neither the polarization nor the spin. It describes a local property in momentum space and is represented in the local representation by the Pauli matrices. This result is achieved by treating the transversality condition on the vector wavefunction as a nonholonomic quantum constraint. We find that the quantum constraint makes it possible to generalize the Stokes parameters to characterize the polarization of a general state. Unexpectedly, the generalized Stokes parameters are specified in a momentum-space local reference system that is fixed by another degree of freedom, called Stratton vector. Only constant Stokes parameters in one particular local reference system can convey the intrinsic degree of freedom of the photon. We show that the optical rotation is one of such processes that change the Stratton vector with the intrinsic quantum number remaining fixed. Changing the Stratton vector of the eigenstate of the helicity will give rise to a Berry's phase.

[12]  arXiv:1803.06733 (cross-list from math.QA) [pdf, ps, other]
Title: Combinatorial bases of principal subspaces of modules for twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$
Subjects: Quantum Algebra (math.QA); Representation Theory (math.RT)

We construct combinatorial bases of principal subspaces of standard modules of level $k \geq 1$ with highest weight $k\Lambda_0$ for the twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$. Using these bases we directly calculate characters of principal subspaces.

[13]  arXiv:1803.06808 (cross-list from math-ph) [pdf, other]
Title: Local martingales associated with SLE with internal symmetry
Authors: Shinji Koshida
Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Probability (math.PR); Quantum Algebra (math.QA); Representation Theory (math.RT)

We consider Schramm-Loewner evolutions with internal degrees of freedom that are associated with representations of affine Lie algebras, following the group theoretical formulation of SLE. We observe that SLEs considered by Bettelheim et al. [PRL 95, 251601 (2005)] and Alekseev et al. [Lett. Math. Phys. 97, 243-261 (2011)] in correlation function formulation are reconstrunced. We also explicitly write down stochastic differential equations on internal degrees of freedom for Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. Our formulation enables to write down several local martingales associated with the solution of SLE from computation on a representation of an affine Lie algebra. Indeed, we write down local martingales associated with solution of SLE for Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. We also find affine $\mathfrak{sl}_{2}$ symmetry of a space of SLE local martingales for the affine $\mathfrak{sl}_{2}$, which can be extended to other affine Lie algebras.

[14]  arXiv:1803.06970 (cross-list from math.DG) [pdf, ps, other]
Title: Higher symmetries of symplectic Dirac operator
Comments: Symplectic Dirac operator, Higher symmetry algebra, Projective differential geometry, Minimal nilpotent orbit, $\mathfrak{sl}(3,\mR)$
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Functional Analysis (math.FA); Representation Theory (math.RT); Symplectic Geometry (math.SG)

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $\mathfrak{sl}(3,{\mathbb{R}})$.

### Replacements for Tue, 20 Mar 18

[15]  arXiv:1609.08593 (replaced) [pdf, other]
Authors: Magdalena Boos
Subjects: Representation Theory (math.RT)
[16]  arXiv:1709.01685 (replaced) [pdf, ps, other]
Title: Regular characters of classical groups over complete discrete valuation rings
Authors: Shai Shechter
Comments: 47 pages. Substantial changes to Sections 2 and 3, following referee report. Some structural changes to Sections 1 and 4 as well
Subjects: Representation Theory (math.RT)
[17]  arXiv:1710.10698 (replaced) [pdf, ps, other]
Title: Extended nilHecke algebra and symmetric functions in type B
Authors: Michael Reeks
Comments: 14 pages. Revised version for publication in Journal of Pure and Applied Algebra. Includes a new section on differentials and DG structure
Subjects: Representation Theory (math.RT)
[18]  arXiv:1801.04738 (replaced) [pdf, ps, other]
Title: Tilting modules over Auslander-Gorenstein Algebras
Subjects: Representation Theory (math.RT)
[19]  arXiv:1604.01938 (replaced) [pdf, ps, other]
Title: On the Noether number of $p$-groups