For any prime p, we construct, and simultaneously count, all of the complex Specht modules in a given p-block of the symmetric group which remain irreducible when reduced modulo p. We call the Specht modules with this property p-irreducible modules. Recently Fayers has proven a conjecture of James and Mathas that provides a characterization of the partitions that correspond to the p-irreducible modules. In this paper we present a method for decomposing the partitions corresponding to p-irreducible modules, and we use this decomposition to construct and count all of the partitions corresponding to p-irreducible Specht modules in a given block.
Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and decompose the result into a direct sum of Specht modules, the irreducible representations of $\mathfrak{S}_t$. We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for $\mathfrak{S}_t$ for $t$ large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternat…
We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations which arises in this context through the geometry of varieties called Springer fibers. We describe a graph that encapsulates combinatorial relations between each of these bases, prove that there is a unique way (up to scaling) to map the Specht basis into the web representatio…
Let H_q(S_n) be the Iwahori-Hecke algebra of the symmetric group. This algebra is semisimple over the rational function field Q(q), where q is an indeterminate, and its irreducible representations over this field are q-analogues S_q(lambda) of the Specht modules of the symmetric group. The q-Specht modules have an "integral form" which is defined over the Laurent polynomial ring Z_[q,q^{-1}] and they come equipped with a natural bilinear form with values in this ring. Now Z[q,q^{-1}] is not a principal ideal domain. Nonetheless, we try to compute the elementary divisors of the Gram matrix of t…
The following thesis contains results on the combinatorial representation theory of the finite Hecke algebra $H_n(q)$. In Chapter 2 simple combinatorial descriptions are given which determine when a Specht module corresponding to a partition $λ$ is irreducible. This is done by extending the results of James and Mathas. These descriptions depend on the crystal of the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_\ell}$. In Chapter 3 these results are extended to determine which irreducible modules have a realization as a Specht module. To do this, a new condition of irr…
The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules $S^λ$. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases -- essentially, only the cases $λ=(d,d)$ and $λ=(d,d,d)$. Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomi…
We construct a new family of homomorphisms from Specht modules into Foulkes modules for the symmetric group. These homomorphisms are used to give a combinatorial description of the minimal partitions (in the dominance order) which label irreducible characters appearing as summands of the characters of Foulkes modules. The homomorphisms are defined using certain families of subsets of the natural numbers. These families are of independent interest; we prove a number of combinatorial results concerning them.
We initiate a study of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group $S_{2n-1}$ on the multilinear component of the free LAnKe with $2n-1$ generators is given by an irreducible representation whose dimension is the $n$th Catalan number. This leads to a more general result on eigenspaces of a certain linear operator, which has additional consequences. We also obtain a new presentation of Specht modules of staircase shape …
Delorme suggested that the set of all complete intersection numerical semigroups can be computed recursively. We have implemented this algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and telescopic numerical semigroups, and numerical semigroups associated to an irreducible plane curve singularity. The recursive nature of this procedure allows us to give bounds for the embedding dimension and for the minimal generators of a semigroup in any of these families.
