# Dr Rowena Paget

Senior Lecturer in Pure Mathematics

Rowena is the School's Equality and Diversity Coordinator and serves on the School's EDI Committee.She joined SMSAS as a lecturer in 2006, having previously worked as a post-doctoral researcher at the University of Leicester and as a college lecturer at the University of Oxford. Her DPhil was in the area of representation theory of symmetric groups at Oxford, under the supervision of Karin Erdmann. Before that she was an undergraduate at Somerville College, Oxford.

## Research interests

Representation theory of groups and algebras, especially symmetric groupsBrauer algebras and other diagram algebras, with a particular interest in plethysm and Foulkes’ Conjecture.Homological algebraAlgebraic combinatorics

## Supervision

Reuben Green (MSc by research: The cellular structure of wreath product algebras, 2015, and currently writing his PhD) Mark Colligan (PhD: Some topics in the representation theory of Brauer algebras, 2012) Melanie de Boeck (PhD: On the structure of Foulkes’ modules for the symmetric group, 2015) Claire Pollard (MSc by research, Foulkes’ Conjecture: current state and computations, 2008) Strong candidates with an interest in the representation theory of symmetric groups or related algebras should contact me by email to discuss possible projects.

## Professional

PGCHE, 2008, University of Kent External examiner for Birkbeck (MSc in Mathematics), from 2017 PhD external examiner (at Oxford, Leeds) Athena SWAN panellist Member of the LMS Outreach: Ri Primary Masterclasses

## Publications

### Forthcoming

• Bowman, C. and Paget, R. (2019). The Uniqueness of Plethystic Factorisation. ArXiv pre-print server [Online]. Available at: https://arxiv.org/pdf/1903.11133.pdf.
We prove that a plethysm product of two Schur functions can be factorised uniquely and classify homogeneous and indecomposable plethysm products.

### Article

• Paget, R. and Wildon, M. (2018). Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions. Proceedings of the London Mathematical Society [Online]. Available at: http://dx.doi.org/10.1112/plms.12210.
This paper proves a combinatorial rule giving all maximal and minimal partitions $\lambda$ such that the Schur function $s_\lambda$ appears in a plethysm of two arbitrary Schur functions. Determining the decomposition of these plethysms has been identified by Stanley as a key open problem in algebraic combinatorics. As corollaries we prove three conjectures of Agaoka on the partitions labeling the lexicographically greatest and least Schur functions appearing in an arbitrary plethysm. We also show that the multiplicity of the Schur function labelled by the lexicographically least constituent may be arbitrarily large. The proof is carried out in the symmetric group and gives an explicit non-zero homomorphism corresponding to each maximal or minimal partition.
• Green, R. and Paget, R. (2017). Iterated Inflations of Cellular Algebras. Journal of Algebra [Online] 493:341-345. Available at: https://doi.org/10.1016/j.jalgebra.2017.09.030.
We present a result characterising iterated inflations of cellular algebras, derived from the work of König and Xi. This result is intended to replace an incorrect proposition in the literature, and gives explicit and readily checked conditions which establish that an algebra is an iterated inflation of cellular algebras, and hence is cellular, with cellular data directly related to the cellular data of the constituent cellular algebras.
• Evseev, A., Paget, R. and Wildon, M. (2014). Character deflations and a generalization of the Murnaghan--Nakayama rule. Journal of Group Theory [Online] 17:1035-1070. Available at: http://www.degruyter.com/view/j/jgth.ahead-of-print/jgth-2014-0023/jgth-2014-0023.xml.
Given natural numbers m and n, we define a deflation map from the characters of the symmetric group S_{mn} to the characters of S_n. This map is obtained by first restricting a character of S_{mn} to the wreath product S_m ?S_n, and then taking the sum of the irreducible constituents of the restricted character on which the base group S_m ×?×S_m acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of S_{mn} under this map. We also prove an analogous result for more general deflation maps in which the base group is not required to act trivially. These results generalize the Murnaghan–Nakayama rule and special cases of the Littlewood–Richardson rule. As a corollary we obtain a new combinatorial formula for the character multiplicities that are the subject of the long-standing Foulkes' Conjecture. Using this formula we verify Foulkes' Conjecture in some new cases.
• Paget, R. and Wildon, M. (2014). Minimal and maximal constituents of twisted Foulkes characters. Journal of the London Mathematical Society [Online repository] 93:301-318. Available at: http://dx.doi.org/10.1112/jlms/jdv070.
We prove combinatorial rules that give the minimal and maximal partitions labelling the irreducible constituents of a family of characters for the symmetric group that generalize Foulkes permutation characters. Restated in the language of symmetric functions, our results determine all minimal and maximal partitions that label Schur functions appearing in the plethysms $s_\nu \circ s_{(m)}$. As a corollary we prove
two conjectures of Agaoka on the lexicographically least constituents of the plethysms $s_\nu \circ s_{(m)}$ and $s_\nu \circ s_{(1^m)}$.
• Paget, R. and Wildon, M. (2011). Set families and Foulkes modules. Journal of Algebraic Combinatorics [Online] 34:525-544. Available at: http://www.springerlink.com/content/t135754v62573165/.
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.
• Paget, R., Hartmann, R., Henke, A. and Koenig, S. (2009). Cohomological stratification of diagram algebras. Mathematische Annalen [Online] 347:765-804. Available at: http://dx.doi.org/10.1007/s00208-009-0458-x.
The class of cellularly stratified algebras is defined and shown to include large classes of diagram algebras. While the definition is in combinatorial terms, by adding extra structure to Graham and Lehrer’s definition of cellular algebras, various structural properties are established in terms of exact functors and stratifications of derived categories. The stratifications relate ‘large’ algebras such as Brauer algebras to ‘smaller’ ones such as group algebras of symmetric groups. Among the applications are relative equivalences of categories extending those found by Hemmer and Nakano and by Hartmann and Paget, as well as identities between decomposition numbers and cohomology groups of ‘large’ and ‘small’ algebras.
• Henke, A. and Paget, R. (2008). Brauer algebras with parameter n = 2 acting on tensor space. Algebras and Representation Theory [Online] 11:545-575. Available at: http://dx.doi.org/10.1007/s10468-008-9092-7.
Let k be a field of prime characteristic p and E an n-dimensional vector space. We completely describe the tensor space E-r viewed as a module for the Brauer algebra B (k) (r,delta) with parameter delta=2 and n=2. This description shows that while the tensor space still affords Schur-Weyl duality, it typically is not filtered by cell modules, and thus will not be equal to a direct sum of Young modules as defined in Hartmann and Paget (Math Z 254:333-357, 2006). This is very different from the situation for group algebras of symmetric groups. Other results about the representation theory of these Brauer algebras are obtained, including a new description of a certain class of irreducible modules in the case when the characteristic is two.
• Paget, R. (2007). A family of modules with Specht and dual Specht filtrations. Journal of Algebra [Online] 312:880-890. Available at: http://dx.doi.org/10.1016/j.jalgebra.2007.03.022.
We study the permutation module arising from the action of the symmetric group S-2n, on the conjugacy class of fixed-point-free involutions, defined over an arbitrary field. The indecomposable direct summands of these modules are shown to possess filtrations by Specht modules and also filtrations by dual Specht modules. We see that these provide counterexamples to a conjecture by Hemmer. Twisted permutation modules are also considered, as is an application to the Brauer algebra.
• Paget, R. (2006). Representation theory of q-rook monoid algebras. Journal of Algebraic Combinatorics [Online] 24:239-252. Available at: http://dx.doi.org/10.1007/s10801-006-0010-y.
We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.
• Hartmann, R. and Paget, R. (2006). Young modules and filtration multiplicities for Brauer algebras. Mathematische Zeitschrift [Online] 254:333-357. Available at: http://dx.doi.org/10.1007/s00209-006-0950-x.
We define permutation modules and Young modules for the Brauer algebra B-k (r, delta), and show that if the characteristic of the field k is neither 2 nor 3 then every permutation module is a sum of Young modules, respecting an ordering condition similar to that for symmetric groups. Moreover, we determine precisely in which cases cell module filtration multiplicities are well-defined, as done by Hemmer and Nakano for symmetric groups.
• Paget, R. (2006). The Mullineux Map for RoCK Blocks. Communications in Algebra [Online] 34:3245-3253. Available at: http://dx.doi.org/10.1080/00927870600778498.
We study the effect of tensoring simple modules in RoCK blocks (also known as Rouquier blocks) of symmetric groups with the one-dimensional sign representation, and the analogue for Hecke algebras of type A. We find an explicit and very neat description. We also describe $p$-regularisation for partitions in these blocks.
• Paget, R. (2005). Induction and decomposition numbers for rock blocks. Quarterly Journal of Mathematics [Online] 56:251-262. Available at: http://dx.doi.org/10.1093/qmath/hah028.
This work is concerned with RoCK blocks (also known as Rouquier blocks) of symmetric groups. A RoCK block, b(rho,w), with abelian defect group is Morita equivalent to a certain block of a wreath product of symmetric group algebras (Chuang and Kessar). Turner specified an idernpotent, e, and conjectured that, for arbitrary weight w, eb(rho,w)e should be Morita equivalent to this block of the wreath product. In this work we provide evidence in support of this conjecture. We prove that the decomposition matrices of these two algebras are identical. As a corollary to the proof, we obtain some knowledge of the composition factors of induced and restricted simple modules.

### Thesis

• Green, R. (2016). The Cellular Structure of Wreath Product Algebras.
We review the definitions and basic theory of cellular algebras as
developed in the papers of Graham and Lehrer and of K ?
onig and Xi.
We then introduce a reformulation of the concept of an iterated inflation
of cellular algebras (a concept due originally to K ?onig and Xi), which
we use to show that the Brauer algebra is cellular (following the work
of K ?onig and Xi). We then review the notion of the wreath product of
an algebra with a symmetric group, and apply our work on iterated
inflations to prove that the wreath product of a cellular algebra with a
symmetric group is in all cases cellular, and we obtain a description of
the cell modules of such a wreath product.
• de Boeck, M. (2015). On the Structure of Foulkes Modules for the Symmetric Group.
This thesis concerns the structure of Foulkes modules for the symmetric group. We study `ordinary' Foulkes modules $H^{(m^n)}$, where $m$ and $n$ are natural numbers, which are permutation modules arising from the action on cosets of $\mathfrak{S}_m\wr\mathfrak{S}_n\leq \mathfrak{S}_{mn}$. We also study a generalisation of these modules $H^{(m^n)}_\nu$, labelled by a partition $\nu$ of $n$, which we call generalised Foulkes modules.

Working over a field of characteristic zero, we investigate the module structure using semistandard homomorphisms. We identify several new relationships between irreducible constituents of $H^{(m^n)}$ and $H^{(m^{n+q})}$, where $q$ is a natural number, and also apply the theory to twisted Foulkes modules, which are labelled by $\nu=(1^n)$, obtaining analogous results.

We make extensive use of character-theoretic techniques to study $\varphi^{(m^n)}_\nu$, the ordinary character afforded by the Foulkes module $H^{(m^n)}_\nu$, and we draw conclusions about near-minimal constituents of $\varphi^{(m^n)}_{(n)}$ in the case where $m$ is even. Further, we prove a recursive formula for computing character multiplicities of any generalised Foulkes character $\varphi^{(m^n)}_\nu$, and we decompose completely the character $\varphi^{(2^n)}_\nu$ in the cases where $\nu$ has either two rows or two columns, or is a hook partition.

Finally, we examine the structure of twisted Foulkes modules in the modular setting. In particular, we answer questions about the structure of $H^{(2^n)}_{(1^n)}$ over fields of prime characteristic.

### Other

• Paget, R. and de Boeck, M. (2014). Decompositions of some twisted Foulkes characters. unspecified [Online repository]. Available at: http://arxiv.org/pdf/1409.0737.pdf.
We decompose the twisted Foulkes characters $\phi^{(2^n)}_\nu$, or equivalently the plethysm $s_\nu \circ s_{(2)}$, in the cases where $\nu$ has either two rows or two columns, or is a hook partition.
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