Macdonald deformation of Vogel's universality and link hyperpolynomials
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918077586735104 |
|---|---|
| author | Bishler, Liudmila Mironov, Andrei Morozov, Alexei |
| author_facet | Bishler, Liudmila Mironov, Andrei Morozov, Alexei |
| contents | Vogel's universality implies a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters $α,β,γ$, which are homogeneous coordinates of Vogel's plane. Actually this is true (if at all) only for a piece of representation theory captured by knot/Chern-Simons theory, where some irreducible representations are often undistinguishable and combined into new ``universally-irreducible" entities (uirreps). We consider from this point of view the recently discovered Macdonald deformation of quantum dimensions, for which a kind of universality holds for the ADE series. The claim is that universal are not Macdonald dimensions themselves, but their products with Littlewood-Richardson coefficients, which themselves are functions of $q$ and $t$ in Macdonald theory. These products are precisely what arises in knot/refined Chern-Simons theory. Actually, we consider the simplest decomposition of adjoint square into six uirreps and obtain the universal formulas for hyperpolynomials of the Hopf link and, more generally, of the torus links $T[2,2n]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Macdonald deformation of Vogel's universality and link hyperpolynomials Bishler, Liudmila Mironov, Andrei Morozov, Alexei High Energy Physics - Theory Mathematical Physics Quantum Algebra Vogel's universality implies a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters $α,β,γ$, which are homogeneous coordinates of Vogel's plane. Actually this is true (if at all) only for a piece of representation theory captured by knot/Chern-Simons theory, where some irreducible representations are often undistinguishable and combined into new ``universally-irreducible" entities (uirreps). We consider from this point of view the recently discovered Macdonald deformation of quantum dimensions, for which a kind of universality holds for the ADE series. The claim is that universal are not Macdonald dimensions themselves, but their products with Littlewood-Richardson coefficients, which themselves are functions of $q$ and $t$ in Macdonald theory. These products are precisely what arises in knot/refined Chern-Simons theory. Actually, we consider the simplest decomposition of adjoint square into six uirreps and obtain the universal formulas for hyperpolynomials of the Hopf link and, more generally, of the torus links $T[2,2n]$. |
| title | Macdonald deformation of Vogel's universality and link hyperpolynomials |
| topic | High Energy Physics - Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2505.16569 |