Instanton counting in Class Sk\mathcal{S}_k

We compute the instanton partition functions of $\mathcal{N}=1$ SCFTs in class $\mathcal{S}_k$. We obtain this result via orbifolding Dp/D(p-4) brane systems and calculating the partition function of the supersymmetric gauge theory on the worldvolume of $K$ D(p-4) branes. Starting with D5/D1 setups probing a $\mathbb{Z}_\ell\times \mathbb{Z}_k$ orbifold singularity we obtain the $K$ instanton partition functions of 6d $(1,0)$ theories on $\mathbb{R}^4 \times T^2$ in the presence of orbifold defects on $T^2$ via computing the 2d superconformal index of the worldvolume theory on $K$ D1 branes wrapping the $T^2$. We then reduce our results to the 5d and to the 4d instanton partition functions. For $k=1$ we check that we reproduce the known elliptic, trigonometric and rational Nekrasov partition functions. Finally, we show that the instanton partition functions of $SU(N)$ quivers in class $\mathcal{S}_k$ can be obtained from the class $\mathcal{S}$ mother theory partition functions with $SU(kN)$ gauge factors via imposing the `orbifold condition' $a_{\mathcal{A}} \rightarrow a_A e^{2\pi i j/k}$ with $\mathcal{A}=jA$ and $A=1,\dots, N$, $j=1,\dots, k$ on the Coulomb moduli and the mass parameters.Comment: 43 pages, 7 figure

Assembling homology classes in automorphism groups of free groups

The observation that a graph of rank $n$ can be assembled from graphs of smaller rank $k$ with $s$ leaves by pairing the leaves together leads to a process for assembling homology classes for $Out(F_n)$ and $Aut(F_n)$ from classes for groups $\Gamma_{k,s}$, where the $\Gamma_{k,s}$ generalize $Out(F_k)=\Gamma_{k,0}$ and $Aut(F_k)=\Gamma_{k,1}$. The symmetric group $\Sigma_s$ acts on $H_*(\Gamma_{k,s})$ by permuting leaves, and for trivial rational coefficients we compute the $\Sigma_s$-module structure on $H_*(\Gamma_{k,s})$ completely for $k \leq 2$. Assembling these classes then produces all the known nontrivial rational homology classes for $Aut(F_n)$ and $Out(F_n)$ with the possible exception of classes for $n=7$ recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of $Aut(F_n)$ and $Out(F_n)$ which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.Comment: Final version for Commentarii Math. Hel

On cocliques in commutative Schurian association schemes of the symmetric group

Given the symmetric group $G = \operatorname{Sym}(n)$ and a multiplicity-free subgroup $H\leq G$, the orbitals of the action of $G$ on $G/H$ by left multiplication induce a commutative association scheme. The irreducible constituents of the permutation character of $G$ acting on $G/H$ are indexed by partitions of $n$ and if $\lambda \vdash n$ is the second largest partition in dominance ordering among these, then the Young subgroup $\operatorname{Sym}(\lambda)$ admits two orbits in its action on $G/H$, which are $\mathcal{S}_\lambda$ and its complement. In their monograph [Erd\H{o}s-Ko-Rado theorems: Algebraic Approaches. {\it Cambridge University Press}, 2016] (Problem~16.13.1), Godsil and Meagher asked whether $\mathcal{S}_\lambda$ is a coclique of a graph in the commutative association scheme arising from the action of $G$ on $G/H$. If such a graph exists, then they also asked whether its smallest eigenvalue is afforded by the $\lambda$-module. In this paper, we initiate the study of this question by taking $\lambda = [n-1,1]$. We show that the answer to this question is affirmative for the pair of groups $\left(G,H\right)$, where $G = \operatorname{Sym}(2k+1)$ and $H = \operatorname{Sym}(2) \wr \operatorname{Sym}(k)$, or $G = \operatorname{Sym}(n)$ and $H$ is one of $\operatorname{Alt}(k) \times \operatorname{Sym}(n-k),\ \operatorname{Alt}(k) \times \operatorname{Alt}(n-k)$, or $\left(\operatorname{Alt}(k)\times \operatorname{Alt}(n-k)\right) \cap \operatorname{Alt}(n)$. For the pair $(G,H) = \left(\operatorname{Sym}(2k),\operatorname{Sym}(k)\wr \operatorname{Sym}(2)\right)$, we also prove that the answer to this question of Godsil and Meagher is negative

Large deviations for random matrices

We prove a large deviation result for a random symmetric n x n matrix with independent identically distributed entries to have a few eigenvalues of size n. If the spectrum S survives when the matrix is rescaled by a factor of n, it can only be the eigenvalues of a Hilbert-Schmidt kernel k(x,y) on [0,1] x [0,1]. The rate function for k is $I(k)=1/2\int h(k(x,y) dxdy$ where h is the Cramer rate function for the common distribution of the entries that is assumed to have a tail decaying faster than any Gaussian. The large deviation for S is then obtained by contraction.Comment: 13 pages. Appeared in Comm. on Stochastic Analysis, vol. 6 no. 1, 1-13, 201

On self-Mullineux and self-conjugate partitions

The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this work, we give an explicit bijection between the two families of partitions in terms of the Mullineux symbol