Combinatorics Seminar 2018

Speaker: Eiichi Bannai (坂内英一), Kyushu University

Date: Tue, Nov 20, 2018

Time: 16:00 - 17:00

Venue: Middle Meeting Room

Title: Unitary $t$-designs and unitary $t$-groups


Unitary $t$-designs are finite subsets of the unitary group $U(d)$ that approximate the whole $U(d)$ well. The positive integer $t$ measures how well they approximate. If a unitary $t$-design itself is a group, then it is called a unitary $t$-group. These concepts were first introduced in physics (quantum information theory). In this talk we first give a quick survey on unitary $t$-designs and unitary $t$-groups. Then we point out that the problem of classifying unitary $t$-groups were studied in the context of finite group theory. We point out that the main parts of the classification that (i) there are no unitary $4$-groups for $U(d)$ for $d\geq 5$ and (ii) unitary $2$-groups are essentially classified for $d\geq 5,$ were done by Guralnick and Tiep (J. of Algebra, 2005), by using the classification of finite simple groups and other deep techniques of finite group theory. Here we give the complete classification of unitary $t$-groups in $U(d)$ for all $t\geq 2$ and $d\geq 2$. It is interesting to note that the classifications for small $d$, say $2\leq d\leq 4$, are closely related to the classification of finite unitary (complex) reflection groups in the sense of Shephard and Todd.

This talk is based on the preprint: Unitary $t$-groups, by Eiichi Bannai, Gabriel Navarro, Noelia Rizo and Pham Huu Tiep, arXiv:1810.02507.

Slides: View slides