A Combinatorial Formula for Graded Multiplicities in Excellent Filtrations

Biswal, Rekha; Kus, Deniz

doi:10.1007/s00031-020-09574-4

Cited by 5 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The case of sl 2 and arbitrary level. A more general approach in the sl 2 case was taken in the articles [12,15] and in the A…”

Section: 5mentioning

confidence: 99%

“…We further discuss the sl 2 case and its connection to the combinatorics of Dyck paths. In [15] a combinatorial formula has been obtained whose ingredients we will now explain. A Dyck path is a diagonal lattice path from the origin (0, 0) to (s, n) for some non-negative integrs s, n ∈ Z + , such that the path never goes below the x-axis.…”

Section: Combinatorics Of Dyck Paths and The Functions A 1→mmentioning

confidence: 99%

“…Even though the study of graded representations of current algebras is mainly motivated by their connection with the representations of quantum affine algebras (via graded limits), they are now of independent interest as they have found many applications in number theory, combinatorics, and mathematical physics. They have connections with mock theta functions, cone theta functions [12,13,15], symmetric Macdonald polynomials [14,27,71], the X = M conjecture [2,59,91], and Schur positivity [55,101] etc. One of the very important families of graded representations of current algebras comes from g-stable Demazure modules.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantum affine algebras, graded limits, and flags

Brito¹,

Chari²,

Kus³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations of these algebras. This connection arises via the notion of the graded and classical limit of finite-dimensional representations of quantum affine algebras. We explain how this study has led to interesting connections with Macdonald polynomials and discuss a BGG-type reciprocity result. We also discuss the role of Demazure modules in this theory and several recent results on the presentation, structure and combinatorics of Demazure modules.

show abstract

“…The case of sl 2 and arbitrary level. A more general approach in the sl 2 case was taken in the articles [12,15] and in the A…”

Section: 5mentioning

confidence: 99%

Section: Combinatorics Of Dyck Paths and The Functions A 1→mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quantum affine algebras, graded limits, and flags

Brito¹,

Chari²,

Kus³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Even though the study of graded representations of current algebras is mainly motivated by their connection with the representations of quantum affine algebras (via graded limits), they are now of independent interest as they have found many applications in number theory, combinatorics, and mathematical physics. They have connections with mock theta functions, cone theta functions 12,13,15 , symmetric Macdonald polynomials 14,26,70 , the X = M conjecture 2,58,90 , and Schur positivity 54,100 etc. One of the very important families of graded representations of current algebras comes from g -stable Demazure modules.…”

Section: Review Articlementioning

confidence: 99%

Quantum Affine Algebras, Graded Limits and Flags

Brito

Chari

Kus³

et al. 2022

J Indian Inst Sci

Self Cite

View full text Add to dashboard Cite

show abstract

“…flags. This approach provides a further deep and unexpected connection to combinatorics and number theory; see for example [5] for the connection to the combinatorics of Dyck path or [2] and [3] for the connection to Mock-Theta functions and hypergeometric series.…”

Section: Introductionmentioning

confidence: 99%

Simplified presentations and embeddings of Demazure modules

Kus¹,

Venkatesh²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

For an untwisted affine Lie algebra we prove an embedding of any higher level Demazure module into a tensor product of lower level Demazure modules (e.g. level one in type A) which becomes in the limit (for anti-dominant weights) the well-known embedding of finite-dimensional irreducible modules of the underlying simple Lie algebra into the tensor product of fundamental modules. To achieve this goal, we first simplify the presentation of these modules extending the results of [13] in the g-stable case. As an application, we propose a crystal theoretic way to find classical decompositions with respect to a maximal semi-simple Lie subalgebra by identifying the Demazure crystal as a connected component in the corresponding tensor product of crystals.

show abstract

A Combinatorial Formula for Graded Multiplicities in Excellent Filtrations

Cited by 5 publications

References 22 publications

Quantum affine algebras, graded limits, and flags

Quantum affine algebras, graded limits, and flags

Quantum Affine Algebras, Graded Limits and Flags

Simplified presentations and embeddings of Demazure modules

Contact Info

Product

Resources

About