Research

https://doi.org/10.1007/s10468-024-10254-0 

The unique factorization theorem, a fundamental result in number theory, asserts that any two multisets of prime numbers yielding the same product must be identical. In 2004, CS Rajan established an analogous result in the context of representation theory, demonstrating that irreducible, integrable representations of finite-dimensional simple Lie algebras exhibit similar uniqueness properties. Notably, this theorem does not generalize to representations of finite groups, as counterexamples can be easily constructed in the representation theory of symmetric groups. 

Subsequent extensions of Rajan's work have been done by Venkatesh and Viswanath (2012) for the class of irreducible, integrable modules of symmetrizable Kac-Moody algebras and by Reif and Venkatesh (2018) for a suitable class of irreducible modules of generalized Kac-Moody algebras (Borcherds Lie algebras). Building upon Rajan's foundational work, our research extends the unique factorization result to a specific class of reducible representations of symmetrizable Kac-Moody algebras called parabolic Verma modules. These subsume the class of all irreducible, integrable representations. We work with the characters of parabolic Verma modules and their restrictions to fixed point subalgebras under Dynkin diagram automorphisms and prove unique factorization result at the level of characters.

https://alco.centre-mersenne.org/articles/10.5802/alco.357/

We define and study a generalization of the Littlewood-Richardson (LR) coefficients, which we call the flagged skew LR coefficients. These subsume several previously studied extensions of the LR coefficients. We establish the saturation property for these coefficients, generalizing the results of Knutson-Tao and Kushwaha-Raghavan-Viswanath. As an intermediate result, we prove that the set of flagged skew tableaux (of a given shape and flag) is a disjoint union of Demazure crystals using the results of Kuono and Assaf-Dranowski-Gonzalez.

https://doi.org/10.48550/arXiv.2306.10289

This paper generalizes a theorem of Littlewood concerning the factorization of Schur polynomials when their variables are twisted by roots of unity. We show that a certain family of flagged skew Schur polynomials admit a similar factorization. These include an interesting family of type-A Demazure characters (key polynomials) as a special case. As an aside, we give a new family of permutations enumerated by the Fuss-Catalan numbers.