arXiv subjects

Takahiro Hayashi

Publications and source records attributed to Takahiro Hayashi.

6 recordsLinked to original sources

arXiv2025

Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins

In this paper, we construct a partial group $\mathcal{P}(F)$ that represents the "partial symmetry" inherent in a subset $F$ of $d$-dimensional Euclidean space. In cases where $F$ is not connected, $\mathcal{P}(F)$ captures more detailed information than the conventional symmetry group $G(F)$. To establish a stronger connection between $\mathcal{P}(F)$ and $F$, we introduce a novel definition of partial group action. Furthermore, to characterize $\mathcal{P}(F)$ in specific cases, we define partial group actions on other partial groups and present a construction of the corresponding semidirect product.

math.GR↗

arXiv2016

Weak Hopf algebras and the distribution of involutions in symmetric groups

By computing Frobenius-Schur indicators of modules of certain weak Hopf algebras, we give a formula for the number of involutions in symmetric groups, which are contained in a given coset with respect to a given Young subgroup.

math.QA↗

arXiv2015

A decomposition rule for certain tensor product representations of the symmetric groups

In this paper, we give a combinatorial rule to calculate the decomposition of the tensor product (Kronecker product) of two irreducible complex representations of the symmetric group ${\mathfrak S}_n$, when one of the representations corresponds to a hook $(n-m, 1^m)$.

math.RT↗

arXiv1999

A canonical Tannaka duality for finite seimisimple tensor categories

For each finite semisimple tensor category, we associate a quantum group (face algebra) whose comodule category is equivalent to the original one, in a simple natural manner. To do this, we also give a generalization of the Tannaka-Krein duality, which assigns a face algebra for each tensor category equipped with an embedding into a certain kind of bimodule category.

math.QA↗

arXiv1999

Coribbon Hopf (face) algebras generated by lattice models

By studying ``points of the underlying quantum groups''of coquasitriangular Hopf (face) algebras, we construct ribbon categories for each lattice models without spectral parameter of both vertex and face type. Also, we give a classification of the braiding and the ribbon structure on quantized classical groups and modular tensor categories closely related to quantum SU(N)_L-invariants of 3-manifolds.

math.QA↗

arXiv1999

Face algebras and unitarity of SU(N)_L-TQFT

Using face algebras (i.e. algebras of L-operators of IRF models), we construct modular tensor categories with positive definite inner product, whose fusion rules and S-matrices are the same as (or slightly different from) those obtained by $U_q (\frak{sl}_{N})$ at roots of unity. Also we obtain state-sums of ABF models on framed links which give quantum SU(2)-invariants of corresponding 3-manifolds.

math.QA↗