Here you can find a brief list of some of the research problems I am interested in pursuing further. Do not hesitate to get in touch if you want to discuss any of these or related questions.
Adjustment matrices.
James's conjecture is fundamental in current research involving decomposition numbers. This is because it predicts that when the weight of a block is smaller than the characteristic of the field, the decomposition matrices for all Hecke algebras are identical to those over complex roots of unity. When the characteristic is smaller or equal to the weight, one can obtain the decomposition matrices by post-multiplying the decomposition matrix over a roots of unity by what is known as the "Adjustment matrix". The conjecture is known to hold for blocks of weight 4 (see Fayers), but no attempt has been made to explicitly compute the adjustment matrices for weight 4 blocks of Hecke algebras when the characteristic of the underlying field is 2 or 3. We expect some of the results used in the computation of the adjustment matrix in weight 3 blocks could be applied or extended to those of weight 4.
Symbolic calculus programs in SageMath.
To the best of my knowledge, there exists no symbolic calculus programs pertaining the modular representation theory of the symmetric group. These could be useful, for example, to explore blocks of arbitrary weight in search for more counterexamples to James's conjecture. At present, the method to find these counterexamples involves the use of Schubert calculus, an area not many representation theorists are acquainted with.
Decomposition numbers via compositions of generalised regularisation maps.
Providing combinatorial formulas for decomposition numbers is one of the key open problems in representation theory of symmetric groups. Computational work allows us to conjecture that the composition of certain regularisation maps on integer partitions provide new decomposition numbers. Attempting to prove this will require further exploration and new ideas.
Cohomology of weight 3 Specht modules.
A few results regarding the Ext-quiver of weight 3 blocks (see for example Fayers and Tan), or the use of cohomological techniques in the study of Specht modules (see Rosas) already exist. I am interested in exploring the extension of some of the results shown by Rosas for the block with empty core in weight 3 to other blocks of the same weight, possibly to all of them. Sadly, my currently poor understanding of homological algebra impedes this from developing.
These are the areas of knowledge in which I have found works applying results from representation theory of symmetric groups. I cite here a few of these as references. I hope to find time at some point to look further into some of these, and perhaps even provide some original contributions!
Game theory [mathematics, behavioural economics].
Representation theory of the symmetric group in voting theory and game theory, K.-D. Crisman and M. Orrison, Cont. Math., Volume 685.
Voting theory [political science].
Voting, the symmetric group, and representation theory, Z. Daugherty et al., American Mathematical Monthly 116 (2009), no. 8, 667-687.
Machine learning [artificial intelligence].
Group theoretical methods in machine learning, R. Kondor, PhD thesis, 2008, Columbia University, London, Canada.
Crystallography [chemistry].
Representation theory for Buckminsterfullerene, G. D. James, J. Algebra, vol 167 (1994), no. 8, 803-820.
Statistical mechanics [physics].
q-Series congruences involving statistical mechanics partition functions in regime III and IV of Baxter’s solution of the hard-hexagon model, M. Merca, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 114 (2020) (In this case the paper uses the theory of integer partitions, rather than that specific to the combinatorial aspects of representations of the symmetric group).
Diagonalisation of the XXZ Hamiltonian by vertex operators, B. Davies et al., Commun. Math. Phys., 151 (1993), 89-153.
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