When Does the Set of (a,b,c)(a, b, c)-Core Partitions Have a Unique Maximal Element?

In 2007, Olsson and Stanton gave an explicit form for the largest $(a, b)$-core partition, for any relatively prime positive integers $a$ and $b$, and asked whether there exists an $(a, b)$-core that contains all other $(a, b)$-cores as subpartitions; this question was answered in the affirmative first by Vandehey and later by Fayers independently. In this paper we investigate a generalization of this question, which was originally posed by Fayers: for what triples of positive integers $(a, b, c)$ does there exist an $(a, b, c)$-core that contains all other $(a, b, c)$-cores as subpartitions? We completely answer this question when $a$, $b$, and $c$ are pairwise relatively prime; we then use this to generalize the result of Olsson and Stanton.Comment: 8 pages, 2 figure

Euler characteristics of Hilbert schemes of points on simple surface singularities

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D.Comment: 57 pages, final version. To appear in European Journal of Mathematic

Foulkes modules and decomposition numbers of the symmetric group

The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in characteristic zero. The main theorem of this paper gives a combinatorial description of certain columns of the decomposition matrices of symmetric groups in odd prime characteristic. The result applies to blocks of arbitrarily high weight. It is obtained by studying the p-local structure of certain twists of the permutation module given by the action of the symmetric group of even degree 2m on the collection of set partitions of a set of size 2m into m sets each of size two. In particular, the vertices of the indecomposable summands of all such modules are characterized; these summands form a new family of indecomposable p-permutation modules for the symmetric group. As a further corollary it is shown that for every natural number w there is a diagonal Cartan number in a block of the symmetric group of weight w equal to w+1

Local-global conjectures and blocks of simple groups

We give an expanded treatment of our lecture series at the 2017 Groups St Andrews conference in Birmingham on local-global conjectures and the block theory of finite reductive groups

Extremal unipotent representations for the finite Howe correspondence

We study the Howe correspondence for unipotent representations of irreducible dual pairs $(G',G)=(\text{U}_m(\mathbb{F}_q),\text{U}_n(\mathbb{F}_q))$ and $(G',G)=(\text{Sp}_{2m}(\mathbb{F}_q),\text{O}^\epsilon_{2n}(\mathbb{F}_q))$, where $\mathbb{F}_q$ denotes the finite field with $q$ elements ($q$ odd) and $\epsilon=\pm 1$. We show how to extract extremal (i.e. minimal and maximal) irreducible subrepresentations from the image of $\pi$ under the correpondence of a unipotent representation $\pi$ of $G$.Comment: 22 page

Results and conjectures on simultaneous core partitions

An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that (2n)- and (2mn+1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type C_n, generalizing a result of Fishel--Vazirani for type A. We also introduce a major statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate simultaneous (2n)- and (2n+1)-core partitions that yield q-analogues of the Coxeter-Catalan numbers of type A and type C. We present related conjectures and open questions on the average size of a simultaneous core partition, q-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q,t-Catalan numbers.Comment: 17 pages; to appear in the European Journal of Combinatoric

Specht modules with abelian vertices

In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily $p^2$-cores where $p$ is the characteristic of the underlying field. Furthermore, in the case of $p\geq 3$, or $p=2$ and $\mu$ is 2-regular, we show that the complexity of the Specht module $S^\mu$ is precisely the $p$-weight of the partition $\mu$. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module $S^{(p^p)}$ for $p\geq 3$