A proof of the $q$-Foulkes conjecture for Gaussian coefficients when $a$ divides $c$
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_ | 1866912515539075072 |
|---|---|
| author | Gutiérrez, Álvaro Szwej, Michał |
| author_facet | Gutiérrez, Álvaro Szwej, Michał |
| contents | Foulkes' conjecture has several generalisations due to Doran, Abdesselam--Chipalkatti, Bergeron, and Troyka. For the special linear Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, these assert that given $a \le c \le d \le b$ with $ab=cd$, the $\mathfrak{sl}_2(\mathbb{C})$-representation $\mathrm{Sym}^a\mathrm{Sym}^b\mathbb{C}^2$ is a subrepresentation of $\mathrm{Sym}^c\mathrm{Sym}^d\mathbb{C}^2$. We present a short proof in the case where $a$ divides $c$ or $d$, which includes all prime values of $a$. This is the first proof in this family of conjectures valid for infinitely many values of $a$; previously only the cases $a=2$ and $a=3$ were known. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A proof of the $q$-Foulkes conjecture for Gaussian coefficients when $a$ divides $c$ Gutiérrez, Álvaro Szwej, Michał Combinatorics Representation Theory 05E10 (Primary) 05E05 20C15 05A30 (Secondary) Foulkes' conjecture has several generalisations due to Doran, Abdesselam--Chipalkatti, Bergeron, and Troyka. For the special linear Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, these assert that given $a \le c \le d \le b$ with $ab=cd$, the $\mathfrak{sl}_2(\mathbb{C})$-representation $\mathrm{Sym}^a\mathrm{Sym}^b\mathbb{C}^2$ is a subrepresentation of $\mathrm{Sym}^c\mathrm{Sym}^d\mathbb{C}^2$. We present a short proof in the case where $a$ divides $c$ or $d$, which includes all prime values of $a$. This is the first proof in this family of conjectures valid for infinitely many values of $a$; previously only the cases $a=2$ and $a=3$ were known. |
| title | A proof of the $q$-Foulkes conjecture for Gaussian coefficients when $a$ divides $c$ |
| topic | Combinatorics Representation Theory 05E10 (Primary) 05E05 20C15 05A30 (Secondary) |
| url | https://arxiv.org/abs/2507.06220 |