SciPost Submission Page
Supersymmetric polynomials and algebro-combinatorial duality
by Dmitry Galakhov, Alexei Morozov, Nikita Tselousov
Submission summary
| Authors (as registered SciPost users): | Dmitry Galakhov |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2407.04810v2 (pdf) |
| Date submitted: | 2024-09-23 20:30 |
| 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
Author comments upon resubmission
First of all we would like to thank the Referees for interesting questions.
Concerning the question of the first referee of the strictness for inequalities (2.2) for half-integer numbers. The reason is that we would treat half-integer numbers as fermions in what follows, so there could not be repetitions of those. However we wanted to keep this section as containing mostly bare definitions one could refer to, and to reveal their meaning in what follows.
All the best, Dmitry.
List of changes
Below are replies for the requests for the minor corrections of the other referee (enumeration coincides with referee's list):
1. Unfortunately, at the moment we are not aware if the super-Schur polynomials represent characters of anything constructive. We added this remark to further problems in concluding Section 6.
2. Absolutely, both quantum toroidal algebras and Yangians contain the algebra they are built on, usually as generators e_k, f_k, ... of lower mode number k. However, usually, for the quantum toroidal algebras this should not be directly the algebra itself rather its quantum version U_q(g). However we did not aim to describe T(\hat gl_{1|1}) from scratch in this text, rather we refer to the definition. We added a more explicit reference to the sources to the preamble of Section 4.
3. Indeed, our construction produces super-Jack and super-Macdonald polynomials similar to the other constructions from the literature.
We added more new relevant references [5-11] including the reference the referee mentioned.
Other modifications in v2:
1. We fixed some typos.
2. We fixed some of author affiliations.