Papers

All of my papers are also available on my arXiv page. Click titles of papers to view them. Alternatively, click 'Abstract' for more details.

In preparation

15. Graded decomposition matrices of cyclotomic quiver Hecke algebras in type C for \(n \leqslant 12\) (with Chris Chung and Andrew Mathas).
    Abstract: (click to expand)

    We compute graded decomposition matrices for the cyclotomic quiver Hecke algebras of affine type \(C\) for \(n\leqslant 12\). These algebras are still very new and very little is known about their graded decomposition numbers. In particular, in contrast to affine type \(A\), we show that graded decomposition numbers of these algebras are not given by the coefficients of the corresponding canonical bases elements, with the first example occurring in type \(C^{(1)}_2\) when \(n=8\) and \(\Lambda = \Lambda_0\).

Preprints

14. Wild blocks of type \(A\) Hecke algebras are strictly wild.
    Abstract: (click to expand)

    We prove that all wild blocks of type \(A\) Hecke algebras with quantum characteristic \(e \geqslant 3\) – i.e. blocks of weight at least 2 – are strictly wild, with the possible exception of the weight 2 Rouquier block for \(e = 3\). As a corollary, we show that for \(e \geqslant 3\), all wild blocks of the q-Schur algebras are strictly wild, without exception.

15. A skew Specht perspective of RoCK blocks and cuspidal systems for KLR algebras in affine type A (with Robert Muth, Thomas Nicewicz, and Louise Sutton).
    Abstract: (click to expand)

    Cuspidal systems parameterize KLR algebra representations via root partitions \(\pi\), where simple modules \(L(\pi)\) arise as heads of proper standard modules. Working in affine type A with an arbitrary convex preorder, we construct explicit skew diagrams \(\zeta(\pi)\) such that the skew Specht module \(S^{\zeta(\pi)}\) has simple head \(L(\pi)\) and a filtration by proper standard modules. This result arises from an in-depth study of the combinatorial interplay between cuspidal systems and RoCK cyclotomic KLR algebras.

Published

1. Schurian-finiteness of blocks of type A Hecke algebras (with Susumu Ariki and Sinéad Lyle). J. Lond. Math. Soc. (2) 108 (2023), no. 6, 2333–2376, DOI.
   Abstract: (click to expand)

   For any algebra \(A\) over an algebraically closed field \(\mathbb{F}\), we say that an \(A\)-module \(M\) is Schurian if \(\mathrm{End}_A(M) \cong \mathbb{F}\). We say that \(A\) is Schurian-finite if there are only finitely many isomorphism classes of Schurian \(A\)-modules, and Schurian-infinite otherwise. By work of Demonet, Iyama and Jasso it is known that Schurian-finiteness is equivalent to \(\tau\)-tilting-finiteness, so that we may draw on a wealth of known results in the subject. We prove that for the type \(A\) Hecke algebras with quantum characteristic \(e\geq 3\), all blocks of weight at least \(2\) are Schurian-infinite in any characteristic. Weight \(0\) and \(1\) blocks are known by results of Erdmann and Nakano to be representation finite, and are therefore Schurian-finite. This means that blocks of type \(A\) Hecke algebras (when \(e\geq 3\)) are Schurian-infinite if and only if they have wild representation type if and only if the module category has finitely many wide subcategories. Along the way, we also prove a graded version of the Scopes equivalence, which is likely to be of independent interest.

2. Decomposable Specht modules indexed by bihooks II (with Robert Muth and Louise Sutton). Algebr. Represent. Theory 26 (2023), no. 1, 241–280, DOI.
   Abstract: (click to expand)

   Previously, the last two authors found large families of decomposable Specht modules labelled by bihooks, over the Iwahori–Hecke algebra of type \(B\). In most cases we conjectured that these were the only decomposable Specht modules labelled by bihooks, proving it in some instances. Inspired by a recent semisimplicity result of Bowman, Bessenrodt and the third author, we look back at our decomposable Specht modules and show that they are often either semisimple, or very close to being so. We obtain their exact structure and composition factors in these cases. In the process, we determine the graded decomposition numbers for almost all of the decomposable Specht modules indexed by bihooks.

3. Strong Gelfand subgroups of \(F\wr S_n\) (with Mahir Can and Yiyang She). Internat. J. Math. 32 (2021), no. 2, 2150010, DOI.
   Abstract: (click to expand)

   The multiplicity-free subgroups (strong Gelfand subgroups) of wreath products are investigated. Various useful reduction arguments are presented. In particular, we show that for every finite group \(F\), the wreath product \(F\wr S_\lambda\), where \(S_\lambda\) is a Young subgroup, is multiplicity-free if and only if \(\lambda\) is a partition with at most two parts, the second part being 0,1, or 2. Furthermore, we classify all multiplicity-free subgroups of hyperoctahedral groups. Along the way, we derive various decomposition formulas for the induced representations from some special subgroups of hyperoctahedral groups.

4. Decomposable Specht modules indexed by bihooks (with Louise Sutton). Pacific J. Math. 304 (2020), no. 2, 655–711, DOI.
   Abstract: (click to expand)

   We study the decomposability of Specht modules labelled by bihooks, bipartitions with a hook in each component, for the Iwahori–Hecke algebra of type \(B\). In all characteristics, we determine a large family of decomposable Specht modules, and conjecture that these provide a complete list of decomposable Specht modules indexed by bihooks. We prove the conjecture for small n.

5. An analogue of row removal for diagrammatic Cherednik algebras (with Chris Bowman). Math. Z. 293 (2019), no. 3, 935–955, DOI.
   Abstract: (click to expand)

   We prove an analogue of James–Donkin row removal theorems for arbitrary diagrammatic Cherednik algebras. This is one of the first results concerning the (graded) decomposition numbers of these algebras over fields of arbitrary characteristic. As a special case, our results yield a new reduction theorem for graded decomposition numbers and extension groups for cyclotomic \(q\)-Schur algebras.

6. Specht modules for quiver Hecke algebras of type C (with Susumu Ariki and Euiyong Park). Publ. Res. Inst. Math. Sci. 55 (2019), no. 3, 565–626, DOI.
   Abstract: (click to expand)

   We construct and investigate Specht modules \(S^\lambda\) for cyclotomic quiver Hecke algebras in type \(C^{(1)}_\ell\) and \(C_\infty\), which are labelled by multipartitions \(\lambda\). It is shown that in type \(C_\infty\), the Specht module \(S^\lambda\) has a homogeneous basis indexed by standard tableaux of shape \(\lambda\), which yields a graded character formula and good properties with the exact functors \(E_i^\Lambda\) and \(F_i^\Lambda\). For type \(C^{(1)}_\ell\), we propose a conjecture.

7. On bases of some simple modules of symmetric groups and Hecke algebras (with Melanie de Boeck, Anton Evseev and Sinéad Lyle). Transform. Groups 23 (2018), no. 3, 631–669, DOI. This paper also refers to some GAP code, available here.
   Abstract: (click to expand)

   We consider simple modules for a Hecke algebra with a parameter of quantum characteristic \(e\). Equivalently, we consider simple modules \(D^\lambda\), labelled by \(e\)-restricted partitions \(\lambda\) of \(n\), for a cyclotomic KLR algebra \(R_n^{\Lambda_0}\) over a field of characteristic \(p\ge 0\), with mild restrictions on \(p\). If all parts of \(\lambda\) are at most \(2\), we identify a set \(\mathrm{DStd}_{e,p}(\lambda)\) of standard \(\lambda\)-tableaux, which is defined combinatorially and naturally labels a basis of \(D^\lambda\). In particular, we prove that the \(q\)-character of \(D^\lambda\) can be described in terms of \(\mathrm{DStd}_{e,p}(\lambda)\). We show that a certain natural approach to constructing a basis of an arbitrary \(D^\lambda\) does not work in general, giving a counterexample to a conjecture of Mathas.

8. On the semisimplicity of the cyclotomic quiver Hecke algebra of type C. Proc. Amer. Math. Soc. 146 (2018), no. 5, 1845–1857, DOI.
   Abstract: (click to expand)

   We provide criteria for the cyclotomic quiver Hecke algebras of type \(C\) to be semisimple. In the semisimple case, we construct the irreducible modules.

9. Kleshchev's decomposition numbers for diagrammatic Cherednik algebras (with Chris Bowman). Trans. Amer. Math. Soc. 370 (2018), no. 5, 3551–3590, DOI.
   Abstract: (click to expand)

   We construct a family of graded isomorphisms between certain subquotients of diagrammatic Cherednik algebras as the quantum characteristic, multicharge, level, degree, and weighting are allowed to vary; this provides new structural information even in the case of the classical \(q\)-Schur algebra. This also allows us to prove some of the first results concerning the (graded) decomposition numbers of these algebras over fields of arbitrary characteristic.

10. A family of graded decomposition numbers for diagrammatic Cherednik algebras (with Chris Bowman and Anton Cox). Int. Math. Res. Not. IMRN 2017 (2017), no. 9, 2686–2734, DOI.
    Abstract: (click to expand)

    We provide an algorithmic description of a family of graded decomposition numbers for diagrammatic Cherednik algebras in terms of affine Kazhdan–Lusztig polynomials.

11. Generalised column removal for graded homomorphisms between Specht modules (with Matthew Fayers). J. Algebraic Combin. 44 (2016), no. 2, 393–432, DOI.
    Abstract: (click to expand)

    Let \(n\) be a positive integer, and let \(\mathscr{H}_n\) denote the affine KLR algebra in type \(A\). Kleshchev, Mathas and Ram have given a homogeneous presentation for graded column Specht modules \(S_\lambda\) for \(\mathscr{H}_n\). Given two multipartitions \(\lambda\) and \(\mu\), we define the notion of a \emph{dominated} homomorphism \(S_\lambda \to S_\mu\), and use the KMR presentation to prove a generalised column-removal theorem for graded dominated homomorphisms between Specht modules. In the process, we prove some useful properties of \(\mathscr{H}_n\)-homomorphisms between Specht modules which lead to an immediate corollary that, subject to a few demonstrably necessary conditions, every homomorphism \(S_\lambda\to S_\mu\) is dominated, and in particular \(\mathrm{Hom}_{\mathscr{H}_n}(S_\lambda,S_\mu)=0\)unless \(\lambda\) dominates \(\mu\). Brundan and Kleshchev show that certain cyclotomic quotients of \(\mathscr{H}_n\) are isomorphic to (degenerate) cyclotomic Hecke algebras of type \(A\). Via this isomorphism, our results can be seen as a broad generalisation of the column-removal results of Fayers and Lyle and of Lyle and Mathas; generalising both into arbitrary level and into the graded setting.

12. Decomposable Specht modules for the Iwahori–Hecke algebra \(\mathscr{H}_{\mathbb{F},-1}(\mathfrak{S}_n)\). J. Algebra 418 (2014), 227–264, DOI.
    Abstract: (click to expand)

    Let \(S_\lambda\) denote the Specht module defined by Dipper and James for the Iwahori–Hecke algebra \(\mathscr{H}_n\) of the symmetric group \(\mathfrak{S}_n\). When \(e=2\) we determine the decomposability of all Specht modules corresponding to hook partitions \((a,1^b)\). We do so by utilising the Brundan–Kleshchev isomorphism between $\mathscr{H}$ and a Khovanov–Lauda–Rouquier algebra and working with the relevant KLR algebra, using the set-up of Kleshchev–Mathas–Ram. When \(n\) is even, we easily arrive at the conclusion that \(S_\lambda\) is indecomposable. When \(n\) is odd, we find an endomorphism of \(S_\lambda\) and use it to obtain a generalised eigenspace decomposition of \(S_\lambda\).