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Supersymmetric polynomials and algebro-combinatorial duality
by Dmitry Galakhov, Alexei Morozov, Nikita Tselousov
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Submission summary
| Authors (as registered SciPost users): | Dmitry Galakhov |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2407.04810v1 (pdf) |
| Date submitted: | 2024-07-12 16:19 |
| Submitted by: | Galakhov, Dmitry |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
In this note we develop a systematic combinatorial definition for constructed earlier supersymmetric polynomial families. These polynomial families generalize canonical Schur, Jack and Macdonald families so that the new polynomials depend on odd Grassmann variables as well. Members of these families are labeled by respective modifications of Young diagrams. We show that the super-Macdonald polynomials form a representation of a super-algebra analog $\mathsf{T}(\widehat{\mathfrak{gl}}_{1|1})$ of Ding-Ioahara-Miki (quantum toroidal) algebra, emerging as a BPS algebra of D-branes on a conifold. A supersymmetric modification for Young tableaux and Kostka numbers are also discussed.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2024-8-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2407.04810v1, delivered 2024-08-19, doi: 10.21468/SciPost.Report.9616
Report
This paper generalizes the canonical Schur, Jack and Macdonald polynomials to their supersymmetric counterparts, with the help of Grassmann variables and newly-defined super-partitions etc.. Especially, with the so-called “algebro-combinatorial duality” introduced, the super-Macdonald polynomials serve as an representation of a DIM super-algebra, the refined version of affine super-Yangian algebra.
It is an important progress relating quantum algebra, which is worthy of more attention. I recommend publication of this nice work.
Just one quick remark: the authors may explain the reason why the sequence inequalities should be strict for odd numbers, below eq. (2.2).
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Anonymous (Referee 2) on 2024-8-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2407.04810v1, delivered 2024-08-18, doi: 10.21468/SciPost.Report.9607
Strengths
1- The work has very strong originality. In particular, a systematic way is presented to construct the super-Macdonald polynomials and Kostka numbers.
2- The algebra T(\hat{gl}_{1|1}) introduced in this article opens up potential applications of the super symmetric polynomials to the string theory and supersymmetric gauge theories.
Weaknesses
1- The article only covers the mathmatical aspects of the super -Macdonald polynomials, and its connection with physics such as the integrable models is to be explored in the future.
Report
The authors presented a systematic way to construct super-Schur, super-Jack, and super-Macdonald symmetric polynomials based on super Young diagrams and the attached Grassmann variables. A trigonometric version of the affine super Yangian is also introduced in the paper and the authors showed that the super Macdonald polynomials give a module of the trigonometric algebra. The article is well-written and organized, and the tools developed in the paper are expected to be applicable in the future study of representation theory and supersymmetric gauge theories. For this reason, I would like to recommend the manuscript be published after clarifying the following minor points.
Requested changes
1. The canonical Schur polynomials (and the skew ones) are deeply related to the representation theory of A-type Lie algebras (and Yangian algebras). Do the super-Schur polynomials also give a certain kind of character to Lie superalgebras/super Yangians?
2. Is there a simple way to see the relation between the trigonometric algebra T(\hat{gl}_{1|1}) and the Lie superalgebra gl_{1|1}? Is it possible to further generalize the algebra to gl_{m|n}?
3. The affine super-Yangian algebra Y(\hat{gl}_{1|1}) is expected to give a supersymmetric Calogero-Sutherland-like model with the super-Jack polynomials being the eigenfunctions. There have been many efforts in the literature to construct supersymmetric extensions of such a model and its eigenbasis, e.g. arXiv:1205.0784. Are the super-Jack polynomials built in this article a completely new version or do they match with some known results in the literature?
Recommendation
Ask for minor revision