Dr Chris Bowman
Before coming to the University, Chris had a research fellowship from Royal Commission for the Exhibition of 1851 at City University of London, and had a postdoc with Eric Vasserot at Paris 7. He completed his PhD at Corpus Christi College, Cambridge in 2011.
Combinatorics and representation theories of diagrammatic algebras (for example symmetric groups, KLR algebras, and partition algebras).
Chris is currently supervising PhD student Dimitris Michailidis who is working on constructing bases of simple modules of KLR algebras.
Bowman, C. and Cox, A. (2018). Modular Decomposition Numbers of Cyclotomic Hecke and Diagrammatic Cherednik Algebras: a Path Theoretic Approach. Forum of Mathematics, Sigma [Online] 6. Available at: https://doi.org/10.1017/fms.2018.9.We introduce a path theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher-level generalizations over fields of arbitrary characteristic. Our first main result is a âsuper-strong linkage principleâ which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalize the notion of homomorphisms between Weyl/Specht modules which are âgenericallyâ placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.
Bowman, C. and Speyer, L. (2018). An analogue of row removal for diagrammatic cherednik algebras. Mathematische Zeitschrift [Online]. Available at: https://doi.org/10.1007/s00209-018-2222-y.We prove an analogue of JamesâDonkin row removal theorems for 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
Bowman, C., Enyang, J. and Goodman, F. (2018). The cellular second fundamental theorem of invariant theory for classical groups. International Mathematics Research Notices [Online]. Available at: https://doi.org/10.1093/imrn/rny079.We construct explicit integral bases for the kernels and the images of diagram algebras (including the symmetric groups, orthogonal and symplectic Brauer algebras) acting on tensor space. We do this by providing an axiomatic framework for studying quotients of diagram algebras.
Bowman, C. and Giannelli, E. (2018). The Integral Isomorphism Behind Row Removal Phenomena For Schur Algebras? Mathematical Proceedings of the Cambridge Philosophical Society [Online]. Available at: https://doi.org/10.1017/S0305004118000294.We explain and generalise row and column removal phenomena for Schur algebras via integral isomorphisms between subquotients of these algebras. In particular, we prove new reduction formulae for p-Kostka numbers.
Bowman, C. and Speyer, L. (2017). Kleshchevâs decomposition numbers for diagrammatic Cherednik algebras. Transactions of the American Mathematical Society [Online] 370:3551-2590. Available at: http://dx.doi.org/10.1090/tran/7054.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.
Bessenrodt, C. and Bowman, C. (2017). Multiplicity-Free Kronecker Products of Characters of the Symmetric Groups. Advances in Mathematics [Online] 322:473-529. Available at: https://doi.org/10.1016/j.aim.2017.10.009.We provide a classification of multiplicity-free inner tensor products of irreducible characters of symmetric groups, thus confirming a conjecture of Bessenrodt. Concurrently, we classify all multiplicity-free inner tensor products of skew characters of the symmetric groups. We also provide formulae for calculating the decomposition of these tensor products.
Bowman, C., De Visscher, D. and Enyang, J. (2017). Simple Modules for the Partition Algebra and Monotone Convergence of Kronecker Coefficients. International Mathematics Research Notices [Online]. Available at: https://doi.org/10.1093/imrn/rnx095.We construct bases of the simple modules for partition algebras which are indexed by paths in an alcove geometry. This allows us to give a concrete interpretation (and new proof) of the monotone convergence property for Kronecker coefficients using stratifications of the cell modules of the partition algebra
Bowman, C., Enyang, J. and Goodman, F. (2017). Diagram algebras, dominance triangularity and skew cell modules. Journal of the Australian Mathematical Society [Online] 104:13-36. Available at: https://doi.org/10.1017/S1446788717000179.We present an abstract framework for the axiomatic study of diagram algebras. Algebras that fit this framework possess analogues of both the Murphy and seminormal bases of the Hecke algebras of the symmetric groups. We show that the transition matrix between these bases is dominance unitriangular. We construct analogues of the skew Specht modules in this setting. This allows us to propose a natural tableaux theoretic framework in which to study the infamous Kronecker problem.
Bowman, C., Speyer, L. and Cox, A. (2016). A family of graded decomposition numbers for diagrammatic Cherednik algebras. A family of graded decomposition numbers for diagrammatic Cherednik algebras [Online]. Available at: http://dx.doi.org/10.1093/imrn/rnw101.We provide an algorithmic description of a family of graded decomposition numbers for diagrammatic Cherednik algebras in terms of affine Kazhdan-Lusztig polynomials.
Bowman, C., De Visscher, M. and King, O. (2015). The Blocks of the Partition Algebra in Positive Characteristic. Algebras and Representation Theory [Online] 18:1357-1388. Available at: https://doi.org/10.1007/s10468-015-9544-9.
Bowman, C., De Visscher, M. and Orellana, R. (2015). The partition algebra and the Kronecker coefficients. Transactions of the American Mathematical Society [Online] 367:3647-3667. Available at: https://doi.org/10.1090/S0002-9947-2014-06245-4.We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. We explain the limiting behaviour and associated bounds in the context of the partition algebra. Our analysis leads to a uniform description of the reduced Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition.
Bowman, C., Doty, S. and Martin, S. (2015). Decomposition of tensor products of modular irreducible representations for SL3: the p ? 5 case. International Electronic Journal of Algebra [Online] 17:105-138. Available at: http://dx.doi.org/10.24330/ieja.266215.We study the structure of the indecomposable direct summands of tensor products of two restricted rational simple modules for the algebraic group SL3(K), where K is an algebraically closed field of characteristic p ? 5. We also give a characteristic-free algorithm for the decomposition of such a tensor product into indecomposable direct summands. The p < 5 case was studied in the authorsâ earlier paper . We find that for characteristics p ? 5 all the indecomposable summands are rigid, in contrast to the characteristic 3 case.
Bowman, C. and Cox, A. (2014). Decomposition numbers for Brauer algebras of type G(m,p,n) in characteristic zero. Journal of Pure and Applied Algebra [Online] 218:992-1002. Available at: https://doi.org/10.1016/j.jpaa.2013.10.014.We introduce Brauer algebras associated to complex reflection groups of type , and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic zero.
Bowman, C. (2013). Bases of quasi-hereditary covers of diagram algebras. Mathematical Proceedings of the Cambridge Philosophical Society [Online] 154:393-418. Available at: https://doi.org/10.1017/S0305004112000667.We extend the the combinatorics of tableaux to the study of Brauer walled Brauer and partition algebras. In particular, we provide uniform constructions of Murphy bases and âSpechtâ filtrations of permutation modules. This allows us to give a uniform construction of semistandard bases of their quasi-hereditary covers.
Bowman, C., Cox, A. and De Visscher, M. (2013). Decomposition numbers for the cyclotomic Brauer algebras in characteristic zero. Journal of Algebra [Online] 378:80-102. Available at: https://doi.org/10.1016/j.jalgebra.2012.12.020.We study the representation theory of the cyclotomic Brauer algebra
via truncation to idempotent subalgebras which are isomorphic
to a product of walled and classical Brauer algebras. In particular,
we determine the block structure and decomposition numbers in
Bowman, C. (2012). Brauer algebras of type C are cellularly stratified. Mathematical Proceedings of the Cambridge Philosophical Society [Online] 153:1-7. Available at: https://doi.org/10.1017/S0305004112000084.In a recent paper Cohen, Liu and Yu introduce the Brauer algebra of type C. We show that this algebra is an iterated inflation of hyperoctahedral groups, and that it is cellularly stratified. This allows us to give an indexing set of the standard modules, results on decomposition numbers, and the conditions under which the algebra is quasi-hereditary.
Bowman, C. and Martin, S. (2012). A Reciprocity Result for Projective Indecomposable Modules of Cellular Algebras and BGG Algebras. Journal of Lie Theory [Online] 22:1065-1073. Available at: http://www.heldermann.de/JLT/JLT22/JLT224/jlt22047.htm.We show that an adaptation of Landrock's Lemma for symmetric algebras also holds for cellular algebras and BGG algebras. This is a result relating the radical layers of any two projective modules. As a corollary we deduce that BGG reciprocity respects Loewy structure.
Bowman, C., Doty, S. and Martin, S. (2011). Decomposition of tensor products of modular irreducible representations for SL3. International Electronic Journal of Algebra [Online] 9:177-219. Available at: http://www.ieja.net/files/papers/volume-9/Volume-8--2010/14-V9-2011.pdf.We give an algorithm for working out the indecomposable direct
summands in a KrullâSchmidt decomposition of a tensor product of two simple
modules for G = SL3 in characteristics 2 and 3. It is shown that there is a
finite family of modules such that every such indecomposable summand is
expressible as a twisted tensor product of members of that family.
Along the way we obtain the submodule structure of various Weyl and
tilting modules. Some of the tilting modules that turn up in characteristic
3 are not rigid; these seem to provide the first example of non-rigid tilting
modules for algebraic groups. These non-rigid tilting modules lead to examples
of non-rigid projective indecomposable modules for Schur algebras, as shown
in the Appendix.
Higher characteristics (for SL3) will be considered in a later paper.