From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys-Murphy elements
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| Format: | Preprint |
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2025
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| _version_ | 1866913875125862400 |
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| author | Coulter, Xavier Do, Norman |
| author_facet | Coulter, Xavier Do, Norman |
| contents | The present work is inspired by three interrelated themes: Weingarten calculus for integration over unitary groups, monotone Hurwitz numbers which enumerate certain factorisations of permutations into transpositions, and Jucys-Murphy elements in the symmetric group algebra. The authors and Moskovsky recently extended this picture to integration on complex Grassmannians, leading to a deformation of the monotone Hurwitz numbers to polynomials that are conjectured to satisfy remarkable interlacing phenomena.
In this paper, we consider integration on the real Grassmannian $\mathrm{Gr}_\mathbb{R}(M,N)$, interpreted as the space of $N \times N$ idempotent real symmetric matrices of rank $M$. We show that in the regime of large $N$ and fixed $\frac{M}{N}$, such integrals have expansions whose coefficients are variants of monotone Hurwitz numbers that are polynomials in the parameter $t = 1 - \frac{N}{M}$.
We define a "$b$-Weingarten calculus", without reference to underlying matrix integrals, that recovers the unitary case at $b = 0$ and the orthogonal case at $b = 1$. The $b$-monotone Hurwitz numbers, previously introduced by Bonzom, Chapuy and Dolega, arise naturally in this context as monotone factorisations of pair partitions. The $b$- and $t$-deformations can be combined to form a common generalisation, leading to the notion of $bt$-monotone Hurwitz numbers, for which we state several results and conjectures.
Finally, we introduce certain linear operators inspired by the aforementioned $b$-Weingarten calculus that can be considered as $b$-deformations of the Jucys-Murphy elements in the symmetric group algebra. We make several conjectures regarding these operators that generalise known properties of the Jucys-Murphy elements and make a connection to the family of Jack symmetric functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_04002 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys-Murphy elements Coulter, Xavier Do, Norman Combinatorics Mathematical Physics Representation Theory 05A15, 05E10, 15B52, 60B20 The present work is inspired by three interrelated themes: Weingarten calculus for integration over unitary groups, monotone Hurwitz numbers which enumerate certain factorisations of permutations into transpositions, and Jucys-Murphy elements in the symmetric group algebra. The authors and Moskovsky recently extended this picture to integration on complex Grassmannians, leading to a deformation of the monotone Hurwitz numbers to polynomials that are conjectured to satisfy remarkable interlacing phenomena. In this paper, we consider integration on the real Grassmannian $\mathrm{Gr}_\mathbb{R}(M,N)$, interpreted as the space of $N \times N$ idempotent real symmetric matrices of rank $M$. We show that in the regime of large $N$ and fixed $\frac{M}{N}$, such integrals have expansions whose coefficients are variants of monotone Hurwitz numbers that are polynomials in the parameter $t = 1 - \frac{N}{M}$. We define a "$b$-Weingarten calculus", without reference to underlying matrix integrals, that recovers the unitary case at $b = 0$ and the orthogonal case at $b = 1$. The $b$-monotone Hurwitz numbers, previously introduced by Bonzom, Chapuy and Dolega, arise naturally in this context as monotone factorisations of pair partitions. The $b$- and $t$-deformations can be combined to form a common generalisation, leading to the notion of $bt$-monotone Hurwitz numbers, for which we state several results and conjectures. Finally, we introduce certain linear operators inspired by the aforementioned $b$-Weingarten calculus that can be considered as $b$-deformations of the Jucys-Murphy elements in the symmetric group algebra. We make several conjectures regarding these operators that generalise known properties of the Jucys-Murphy elements and make a connection to the family of Jack symmetric functions. |
| title | From Weingarten calculus for real Grassmannians to deformations of monotone Hurwitz numbers and Jucys-Murphy elements |
| topic | Combinatorics Mathematical Physics Representation Theory 05A15, 05E10, 15B52, 60B20 |
| url | https://arxiv.org/abs/2506.04002 |