L.Yu. Tsiovkina. Some Schurian association schemes related to Suzuki and Ree groups ... P. 249-254

An association scheme is a pair $(\Omega,\mathcal{R})$ consisting of a finite set $\Omega$ and a set $\mathcal{R}=\{R_0,R_1\ldots, R_s\}$ of binary relations on $\Omega$ satisfying the following conditions: (1) $\mathcal{R}$ is a partition of the set $\Omega^2$; (2) $\{(x,x)\ |\ x\in \Omega\}\in \mathcal{R}$; (3) ${R_t}^T=\{(y,x)\ |\ (x,y)\in R_t\}\in {\mathcal R}$ for all $0\le t\le s$; (4) for all $0\le i,j,t\le s$, there exist constants $c_{ij}^t$ (called the intersection numbers of the scheme) such that $c_{ij}^t=|\{z\in \Omega| (x,z)\in R_i, (z,y)\in R_j\}|$ for any pair $(x,y)\in R_t$. An association scheme $(\Omega,\mathcal{R})$ is called Schurian if, for some permutation group on $\Omega$, the set of orbitals of this group on $\Omega$ coincides with $\mathcal{R}$. This work is devoted to the study of Schurian association schemes related to Suzuki groups $Sz(q)$ and Ree groups ${^2G}_2(q)$ with $q>3$ for which some graphs of their basic relations are antipodal distance-regular graphs of diameter 3. Assume that $G$ is one of the mentioned groups, $r=(q-1)_{2'}$, $B$ is a Borel subgroup of $G$, $U$ is a unipotent subgroup of $G$ contained in $B$, $K$ is a subgroup of $B$ with index $r$, $g$ is an involution in $G-B$, and $f$ is an element of order $r$ in $B\cap B^g$. Let$\Omega$ be the set of the right $K$-cosets of $G$, and put $h_i=f^i$ and $h_{r+i}=gf^i$ for all $i\in \{0,\ldots,r-1\}$. Denote by ${\mathcal{R}}$ the set $\{R_0,R_1,\ldots, R_{2r-1}\}$ of binary relations on $\Omega$ defined for each $t\in \{0,1,\ldots,2r-1\}$ by the rule: $(Kx,Ky)\in R_t$ if and only if $xy^{-1}$ is contained in the double coset $Kh_tK$. We prove that ${\mathcal X}=(\Omega, {\mathcal{R}})$ is a Schurian association scheme and its set of basic relations coincides with the set of orbitals of $G$ on $\Omega$. We find that the intersection number $c_{ij}^t$, where $0\le i,j,t\le 2r-1$, of the scheme ${\mathcal X}$ is $|U|$ if $t\le r-1$, $i,j\ge r$, and $j-i\equiv t \pmod r$; $(|U|-1)/r$ if $ i,j,t\ge r$; 1 if either $t\le r-1$, $i,j\le r-1$, and $ i+j\equiv t \pmod r$, or $i\le r-1$, $t,j\ge r$, and $ j-i\equiv t \pmod r$, or $t,i\ge r$, $j\le r-1$, and $ i+j\equiv t \pmod r$; and 0 in the remaining cases, where $|U|=q^2$ if $G=Sz(q)$ and $|U|=q^3$ if $G={^2G}_2(q)$. As a corollary, we find the structural parameters $m_{h_t}(h_i,h_j)=|\{Kx\in \Omega |\ Kx\subseteq Kh_i^{-1}Kh_t\cap Kh_jK\}|$ of the Hecke algebra $\mathbb{C}(K{\setminus}G/K)$ of $G$ with respect to $K$. Namely, we show that $m_{h_t}(h_i,h_j)$ is exactly the intersection number $c_{ij}^t$ of the scheme ${\mathcal X}$ for all $0\le i,j,t\le 2r-1$. By definition, the graph of the basic relation $R_t$ with $t\ge r$ of ${\mathcal X}$ is equivalent to the coset graph $\Gamma(G,K,Kh_tK)$ of $G$ with respect to $K$ and the element $h_t$ and, as is known, is an antipodal distance-regular graph of diameter 3 with intersection array $\{|U|,(|U|-1)(r-1)/r,1;1,(|U|-1)/r,|U|\}$. The latter fact was proved in the author's earlier paper, where we proposed a technique for studying\linebreak the graphs $\Gamma(G,K,Kh_tK)$; the technique is based on analyzing the mutual distribution of the neighborhoods of vertices. In the present work, we prove the distance regularity of these graphs as a corollary of the properties of the scheme ${\mathcal X}$.

Keywords: Schurian association scheme, distance-regular graph, antipodal graph

Received September 5, 2019

Revised October 23, 2019

Accepted October 28, 2019

Lyudmila Yuryevna Tsiovkina, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: l.tsiovkina@gmail.com

REFERENCES

1.   Muzychuk M., Ponomarenko I. Association schemes, Schur rings and the isomorphism problem for circulant graphs, Part 1, Notes of lectures given at the international workshop “Algorithmic problems in group theory and related areas”, Novosibirsk, 2014, pp. 1–24. Available at: http://www.math.nsc.ru/conference/isc/2014/lectures/MP1_2014.pdf .

2.   Tsiovkina L.Yu. Two new infinite families of arc-transitive antipodal distance-regular graphs of diameter three with $\lambda =\mu $ related to groups $Sz(q)$ and ${^2}G_2(q)$. J. Algebr. Comb., 2015, vol. 41, no. 4, pp. 1079–1087. doi: 10.1007/s10801-014-0566-x .

3.   Aschbacher M. Finite group theory. Cambridge: Cambridge University Press, 2000, 305 p. ISBN: 0-521-78675-4 .

4.   Carter R.W. Simple groups of Lie type. London: Wiley, 1972, 332 p. ISBN: 0471137359

Cite this article as: L.Yu.Tsiovkina. Some Schurian association schemes related to Suzuki and Ree groups, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 249–254.