Fused Hecke algebra and one-boundary algebras
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arXiv
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| Format: | Preprint |
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2023
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| author | d'Andecy, L. Poulain Zaimi, M. |
| author_facet | d'Andecy, L. Poulain Zaimi, M. |
| contents | This paper gives an algebraic presentation of the fused Hecke algebra which describes the centraliser of tensor products of the $U_q(gl_N)$-representation labelled by a one-row partition of any size with vector representations. It is obtained through a detailed study of a new algebra that we call the symmetric one-boundary Hecke algebra. In particular, we prove that the symmetric one-boundary Hecke algebra is free over a ring of Laurent polynomials in three variables and we provide a basis indexed by a certain subset of signed permutations. We show how the symmetric one-boundary Hecke algebra admits the one-boundary Temperley-Lieb algebra as a quotient, and we also describe a basis of this latter algebra combinatorially in terms of signed permutations with avoiding patterns. The quotients corresponding to any value of $N$ in $gl_N$ (the Temperley-Lieb one corresponds to $N=2$) are also introduced. Finally, we obtain the fused Hecke algebra, and in turn the centralisers for any value of $N$, by specialising and quotienting the symmetric one-boundary Hecke algebra. In particular, this generalises to the Hecke case the description of the so-called boundary seam algebra, which is then obtained (taking $N=2$) as a quotient of the fused Hecke algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_10937 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fused Hecke algebra and one-boundary algebras d'Andecy, L. Poulain Zaimi, M. Representation Theory Mathematical Physics Quantum Algebra This paper gives an algebraic presentation of the fused Hecke algebra which describes the centraliser of tensor products of the $U_q(gl_N)$-representation labelled by a one-row partition of any size with vector representations. It is obtained through a detailed study of a new algebra that we call the symmetric one-boundary Hecke algebra. In particular, we prove that the symmetric one-boundary Hecke algebra is free over a ring of Laurent polynomials in three variables and we provide a basis indexed by a certain subset of signed permutations. We show how the symmetric one-boundary Hecke algebra admits the one-boundary Temperley-Lieb algebra as a quotient, and we also describe a basis of this latter algebra combinatorially in terms of signed permutations with avoiding patterns. The quotients corresponding to any value of $N$ in $gl_N$ (the Temperley-Lieb one corresponds to $N=2$) are also introduced. Finally, we obtain the fused Hecke algebra, and in turn the centralisers for any value of $N$, by specialising and quotienting the symmetric one-boundary Hecke algebra. In particular, this generalises to the Hecke case the description of the so-called boundary seam algebra, which is then obtained (taking $N=2$) as a quotient of the fused Hecke algebra. |
| title | Fused Hecke algebra and one-boundary algebras |
| topic | Representation Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2306.10937 |