arXiv subjects
Publications and source records attributed to Eoghan McDowell.
We classify when a spin and a non-spin irreducible character of the alternating group have proportional 2-modular reductions. Equivalently, we classify when such a pair of characters are proportional on elements of odd order.
Let $G$ be a finite group and $p$ a prime. It is interesting to determine when two ordinary irreducible representations of $G$ have the same $p$-modular reduction; this is the same as saying that the corresponding rows of the decomposition matrix are equal, or that the characters of the two representations agree on $p$-regular conjugacy classes. In fact we consider the more general problem of asking when two rows of the decomposition matrix are proportional. In the case where $G$ is a double cover of the alternating or symmetric group, this problem has been solved except when $p=3$. Here we resolve the missing case for spin characters (i.e. characters which are not lifted from the covered group), which completely solves the problem for the double cover of the symmetric group. There are surprising parallels to our solution to the corresponding problem for $p=2$.
For a finite group, it is interesting to determine when two ordinary irreducible representations have the same $p$-modular reduction; that is, when two rows of the decomposition matrix in characteristic $p$ are equal, or equivalently when the corresponding $p$-modular Brauer characters are the same. We complete this task for the double covers of the symmetric group when $p=2$, by determining when the $2$-modular reduction of an irreducible spin representation coincides with a $2$-modular Specht module. In fact, we obtain a more general result: we determine when an irreducible spin representation has $2$-modular Brauer character proportional to that of a Specht module. In the course of the proof, we use induction and restriction functors to construct a function on generalised characters which has the effect of swapping runners in abacus displays for the labelling partitions.
The Weyl modules are the standard modules for the Schur algebra. Their duals (the costandard modules) have well-known constructions as quotients of exterior powers and as submodules of symmetric powers. This paper presents analogous constructions for the Weyl modules themselves, introducing relations that are a non-trivial dualisation of the Garnir relations. More generally, our constructions describe endofunctors on the category of representations of any group.
We show that the $p$-part of the degree of an irreducible character of a symmetric group is completely determined by the set of vanishing elements of $p$-power order. As a corollary we deduce that the set of zeros of prime power order controls the degree of such a character. The same problem is analysed for alternating groups, where we show that when $p=2$ this data can only be determined up to two possibilities. We prove analogous statements for the defect of the $p$-block containing the character and for the $p$-height of the character.
This paper identifies all pairs of ordinary irreducible characters of the alternating group which agree on conjugacy classes of elements of order not divisible by a fixed integer $l$, for $l \neq 3$. We do the same for the double covers of the symmetric and alternating groups. The only such characters are the conjugate or associate pairs labelled by partitions with a certain parameter divisible by $l$. When $l$ is prime, this implies that the rows of the $l$-modular decomposition matrix are distinct except for the rows labelled by these pairs. When $l=3$ we exhibit many additional examples of such pairs of characters.
This paper gives a necessary and sufficient condition for the image of the Specht module under the inverse Schur functor to be isomorphic to the dual Weyl module in characteristic 2, and gives an elementary proof that this isomorphism holds in all cases in all other characteristics. These results are new in characteristics 2 and 3. We deduce some new examples of indecomposable Specht modules in characteristic 2. When the isomorphism does not hold, the dual Weyl module is still a quotient of the image of the Specht module, and we prove some additional results: we demonstrate that the image need not have a filtration by dual Weyl modules, we bound the dimension of the kernel of the quotient map, and we give some explicit descriptions for particular cases. Our method is to view the Specht and dual Weyl modules as quotients of suitable exterior powers by the Garnir relations.
This paper proves an identity between flagged Schur polynomials, giving a duality between row flags and column flags. This identity generalises both the binomial determinant duality theorem due to Gessel and Viennot and the symmetric function duality theorem due to Aitken. As corollaries we obtain the lifts of the binomial determinant duality theorem to $q$-binomial coefficients and to symmetric polynomials. Our method is a path counting argument on a novel lattice generalising that used by Gessel and Viennot.
This paper investigates partitions which have neither parts nor hook lengths divisible by $p$, referred to as $p$-core $p'$-partitions. We show that the largest $p$-core $p'$-partition corresponds to the longest walk on a graph with vertices $\{0, 1, \ldots, p-1\}$ and labelled edges defined via addition modulo $p$. We also exhibit an explicit family of large $p$-core $p'$-partitions, giving a lower bound on the size of the largest such partition which is of the same degree as the upper bound found by McSpirit and Ono.
Let $E$ be the natural representation of the special linear group $\mathrm{SL}_2(K)$ over an arbitrary field $K$. We use the two dual constructions of the symmetric power when $K$ has prime characteristic to construct an explicit isomorphism $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \mathrm{Sym}_\ell \mathrm{Sym}^m E$. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \bigwedge^m \mathrm{Sym}^{\ell+m-1} E$. We also generalise a result first proved by King, by showing that if $\nabla^λ$ is the Schur functor for the partition $λ$ and $λ^\circ$ is the complement of $λ$ in a rectangle with $\ell+1$ rows, then $\nabla^λ\mathrm{Sym}^\ell E \cong \nabla^{λ^\circ} \mathrm{Sym}_\ell E$. To illustrate that the existence of such `plethystic isomorphisms' is far from obvious, we end by proving that the generalisation $\nabla^λ\mathrm{Sym}^\ell E \cong \nabla^{λ'} \mathrm{Sym}^{\ell + \ell(λ') - \ell(λ)}E$ of the Wronskian isomorphism, known to hold for a large class of partitions over the complex field, does not generalise to fields of prime characteristic, even after considering all possible dualities.
In this paper we introduce a novel family of Markov chains on the simple representations of $\mathrm{SL}_2(\mathbb{F}_p)$ in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of $\mathrm{SL}_2(\mathbb{F}_p)$, emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple $\mathrm{SL}_2(\mathbb{F}_p)$-representations.