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Mai Katada

Publications and source records attributed to Mai Katada.

10 recordsLinked to original sources
arXiv2026

A family of generalized Gelfand pairs attached to $3$-step nilpotent Lie groups

It is well known that if $(K\ltimes N, K)$ is a Gelfand pair with $N$ a nilpotent Lie group and $K$ a compact subgroup of the automorphism group of $N$, then $N$ is at most $2$-step. The notion of Gelfand pairs is extended to the notion of generalized Gelfand pairs, where $K$ is not necessarily compact. Gallo and Saal constructed an example of a generalized Gelfand pair $(K\ltimes N, K)$ with $N$ $3$-step and $K$ non-compact. In this work, we construct a family of generalized Gelfand pairs $(K_d \ltimes N_d, K_d)_{d\ge 1}$, where $N_d$ is $3$-step and $K_d$ is isomorphic to $\mathbb{R}^{d+1}$; the case of $d=1$ recovers the example of Gallo and Saal.

math.FA↗
arXiv2025

Extensions between functors from Jacobi diagrams in handlebodies

The first Ext-groups between Schur functors in the category of modules over the $\Bbbk$-linearization $\Bbbk\mathbf{gr}^{\operatorname{op}}$ of the opposite of the category of finitely generated free groups are computed for a filed $\Bbbk$ of characteristic $0$. The $\Bbbk$-linear category $\mathbf{A}$ of Jacobi diagrams in handlebodies, which was introduced by Habiro and Massuyeau, has an $\mathbb{N}$-grading whose degree $0$ part identifies with the category $\Bbbk\mathbf{gr}^{\operatorname{op}}$. We compute the first Ext-groups in the category of $\mathbf{A}$-modules between simple $\mathbf{A}$-modules which are induced by Schur functors.

math.CT↗
arXiv2025

Modules over the category of Jacobi diagrams in handlebodies

The linear category $\mathbf{A}$ of Jacobi diagrams in handlebodies was introduced by Habiro and Massuyeau. We study the category of modules over the category $\mathbf{A}$. We generalize the adjunction given by Powell to an adjunction between the category of $\mathbf{A}$-modules and the category of modules over the linear PROP for Casimir Lie algebras by using the category $\mathbf{A}^{L}$ of extended Jacobi diagrams in handlebodies. Then we study subquotient $\mathbf{A}$-modules of $\mathbf{A}(0,-)$.

math.RT↗
arXiv2024

The stable Albanese homology of the IA-automorphism groups of free groups

The IA-automorphism group $\operatorname{IA}_n$ of the free group $F_n$ of rank $n$ is a normal subgroup of the automorphism group $\operatorname{Aut}(F_n)$ of $F_n$. We study the Albanese homology of $\operatorname{IA}_n$, which is the quotient of the rational homology of $\operatorname{IA}_n$ defined as the image of the map induced by the abelianization map of $\operatorname{IA}_n$ on homology. The Albanese homology of $\operatorname{IA}_n$ is an algebraic $\operatorname{GL}(n,\mathbb{Q})$-representation. We determine the representation structure of the Albanese homology of $\operatorname{IA}_n$ for $n$ greater than or equal to three times the homological degree. We also determine the structure of the stable Albanese homology of the analogue of $\operatorname{IA}_n$ to the outer automorphism group of $F_n$. Moreover, we identify the relation between the stable Albanese (co)homology of $\operatorname{IA}_n$ and the stable cohomology of $\operatorname{Aut}(F_n)$ with certain twisted coefficients.

math.AT↗
arXiv2024

On the stable cohomology of the IA-automorphism groups of free groups

Borel's stability and vanishing theorem gives the stable cohomology of $\mathrm{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations. By combining the Borel theorem with the Hochschild-Serre spectral sequence, we compute the twisted first cohomology of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$. We also study the stable rational cohomology of the IA-automorphism group $\mathrm{IA}_n$ of $F_n$. We propose a conjectural algebraic structure of the stable rational cohomology of $\mathrm{IA}_n$, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces.

math.AT↗
arXiv2022

On Borel's stable range of the twisted cohomology of $\mathrm{GL}(n,\mathbb{Z})$

Borel's stability and vanishing theorem gives the stable cohomology of $\mathrm{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations. We compute the improved stable range that Borel remarked about. In order to further improve Borel's stable range, we adapt the method of Kupers-Miller-Patzt to any algebraic $\mathrm{GL}(n,\mathbb{Z})$-representations.

math.AT↗
arXiv2022

Stable rational homology of the IA-automorphism groups of free groups

The rational homology of the IA-automorphism group $\operatorname{IA}_n$ of the free group $F_n$ is still mysterious. We study the quotient of the rational homology of $\operatorname{IA}_n$ that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of $\operatorname{IA}_n$. We obtain a representation-stable $\operatorname{GL}(n,\mathbb{Q})$-subquotient of the Albanese homology of $\operatorname{IA}_n$, which conjecturally coincides with the entire Albanese homology of $\operatorname{IA}_n$. In particular, we obtain a lower bound of the dimension of the Albanese homology of $\operatorname{IA}_n$ for each homological degree in a stable range. Moreover, we determine the entire third Albanese homology of $\operatorname{IA}_n$ for $n\ge 9$. We also study the Albanese homology of an analogue of $\operatorname{IA}_n$ to the outer automorphism group of $F_n$ and the Albanese homology of the Torelli groups of surfaces. Moreover, we study the relation between the Albanese homology of $\operatorname{IA}_n$ and the cohomology of $\operatorname{Aut}(F_n)$ with twisted coefficients.

math.AT↗
arXiv2021

Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I

We consider an action of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$ on the filtered vector space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. This action induces on the associated graded vector space of $A_d(n)$, which is identified with the space $B_d(n)$ of open Jacobi diagrams, an action of the general linear group $\mathrm{GL}(n,Z)$ and an action of the graded Lie algebra of the IA-automorphism group of $F_n$ associated with its lower central series. We use these actions on $B_d(n)$ to study the $\mathrm{Aut}(F_n)$-module structure of $A_d(n)$. In particular, we consider the case where $d=2$ in detail and give an indecomposable decomposition of $A_2(n)$. We also construct a polynomial functor $A_d$ of degree $2d$ from the opposite category of the category of finitely generated free groups to the category of filtered vector spaces, which includes the $\mathrm{Aut}(F_n)$-module structure of $A_d(n)$ for all $n\geq 0$.

math.QA↗
arXiv2021

Actions of automorphism groups of free groups on spaces of Jacobi diagrams. II

The automorphism group $\operatorname{Aut}(F_n)$ of the free group $F_n$ acts on a space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. We study the $\operatorname{Aut}(F_n)$-module structure of $A_d(n)$ by using two actions on the associated graded vector space of $A_d(n)$: an action of the general linear group $\operatorname{GL}(n,Z)$ and an action of the graded Lie algebra $\mathrm{gr}(\operatorname{IA}(n))$ of the IA-automorphism group $\operatorname{IA}(n)$ of $F_n$ associated with its lower central series. We extend the action of $\mathrm{gr}(\operatorname{IA}(n))$ to an action of the associated graded Lie algebra of the Andreadakis filtration of the endomorphism monoid of $F_n$. By using this action, we study the $\operatorname{Aut}(F_n)$-module structure of $A_d(n)$. We obtain an indecomposable decomposition of $A_d(n)$ as $\operatorname{Aut}(F_n)$-modules for $n\geq 2d$. Moreover, we obtain the radical filtration of $A_d(n)$ for $n\geq 2d$ and the socle of $A_3(n)$.

math.GT↗