Components of $V(mρ) \otimes V(nρ)$
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| Format: | Preprint |
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2026
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| _version_ | 1866914614169567232 |
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| author | Biswal, Rekha Jeralds, Sam |
| author_facet | Biswal, Rekha Jeralds, Sam |
| contents | Let $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra and let $ρ$ denote the sum of the fundamental weights. The irreducible highest weight representations $V(mρ)$ occupy a distinguished position in representation theory due to their rich symmetry and geometric significance. In this paper, we study the tensor products \[ V(mρ)\otimes V(nρ), \quad m,n \in \mathbb{N}, \] and investigate the structure of their irreducible decompositions. Motivated by the classical conjecture of Kostant, which predicts a highly structured behavior in simpler settings, we propose a general framework describing the irreducible components appearing in such tensor products for finite-dimensional semisimple or affine Kac-Moody Lie algebras $\mathfrak{g}$. Our results identify a family of dominant weights governing the decomposition and provide criteria for their occurrence. This work extends the scope of Kostant-type phenomena and reveals new structural patterns in tensor products associated with multiples of the Weyl vector. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_29802 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Components of $V(mρ) \otimes V(nρ)$ Biswal, Rekha Jeralds, Sam Representation Theory 17B10, 17B67, 17B22, 05E10 Let $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra and let $ρ$ denote the sum of the fundamental weights. The irreducible highest weight representations $V(mρ)$ occupy a distinguished position in representation theory due to their rich symmetry and geometric significance. In this paper, we study the tensor products \[ V(mρ)\otimes V(nρ), \quad m,n \in \mathbb{N}, \] and investigate the structure of their irreducible decompositions. Motivated by the classical conjecture of Kostant, which predicts a highly structured behavior in simpler settings, we propose a general framework describing the irreducible components appearing in such tensor products for finite-dimensional semisimple or affine Kac-Moody Lie algebras $\mathfrak{g}$. Our results identify a family of dominant weights governing the decomposition and provide criteria for their occurrence. This work extends the scope of Kostant-type phenomena and reveals new structural patterns in tensor products associated with multiples of the Weyl vector. |
| title | Components of $V(mρ) \otimes V(nρ)$ |
| topic | Representation Theory 17B10, 17B67, 17B22, 05E10 |
| url | https://arxiv.org/abs/2605.29802 |