For a group \(G\) and a subgroup \(H\subseteq G\), consider the coset action \(\sigma : G \longrightarrow S_{G/H}\), on the coset space \(G/H\). This produces a complex representation \(\widetilde{\sigma}\) of \(G\). The pair \((G,H)\) is called to be a Gelfand pair if the centralizer algebra \(C(\widetilde{\sigma})\) is commutative. This is equivalent to saying that the induction of the trivial character of \(H\) gives a multiplicity-free character of \(G\). A pair \((G,H)\) of finite groups \(G\), \(H\) with \(H\subseteq G\) is called a strong Gelfand pair if every irreducible character of \(H\) induces a multiplicity-free character of \(G\). Note that a strong Gelfand pair is also referred to as “multiplicity one property” or “multiplicity one theorem”.
This article determines the strong Gelfand pairs when \(G=\mathrm{SL}(2,p)\), the special linear group of \(2\times 2\) matrices over the field of prime order \(p\). Consider the following two subgroups of \(\mathrm{SL}(2,p)\). The first one is \(\mathrm{U}(2,p)\), the subgroup of upper triangular matrices in \(\mathrm{SL}(2,p)\) and the second one being \(T\), the unique subgroup of index \(2\), of \(\mathrm{U}(2,p)\). Then the main result of the paper says that for a prime \(p>11\),

●

if \(p\equiv 1\pmod{4}\), then the only strong Gelfand pair is given by \(\left(\mathrm{SL}(2,p),\mathrm{U}(2,p)\right)\),

●

if \(p\equiv 3\pmod{4}\), then there are exactly two strong Gelfand pairs: \(\left(\mathrm{SL}(2,p),\mathrm{U}(2,p)\right)\) and \(\left(\mathrm{SL}(2,p),T\right)\).

The scenario for \(p\leq 11\) is a little different. We have the following cases.

●

For \(p=2\), \(\mathrm{SL}(2,2)\) is isomorphic to the symmetric group on \(3\) letters. The strong Gelfand pairs are \((\mathrm{SL}(2,2), C_2)\) and \((\mathrm{SL}(2,2), C_3)\).

●

For \(p=3\), the strong Gelfand pairs are \((\mathrm{SL}(2,3),C_3)\), \((\mathrm{SL}(2,3),C_6)\) and \((\mathrm{SL}(2,3),\mathrm{U}(2,3))\).

●

For \(p=5\), the strong Gelfand pairs are \((\mathrm{SL}(2,5),\mathrm{SL}(2,3))\) and \((\mathrm{SL}(2,5),\mathrm{U}(2,5))\).

●

For \(p=7\), the strong Gelfand pairs are \((\mathrm{SL}(2,7),\mathrm{U}(2,7))\), \((\mathrm{SL}(2,7), T)\) and \((\mathrm{SL}(2,7), K)\) where \(K\) is the preimage of \(\Sigma_4\subseteq\mathrm{PSL}(2,7)\) (the symmetric group on \(4\) symbols) under the canonical projection map.

●

For \(p=11\), the strong Gelfand pairs are \((\mathrm{SL}(2,11),\mathrm{U}(2,11))\), \((\mathrm{SL}(2,11), T)\) and \((\mathrm{SL}(2,11), 2I)\) where \(2I=\langle r,s,t|r^2=s^3=t^5=rst \rangle\).

The cases \(p\leq 11\) have been computed using Magma. For the rest of the cases, the approach has been uniform, as follows. The authors start with the description of character tables of the groups \(\mathrm{SL}(2,p)\), \(\mathrm{U}(2,p)\) and the projective special linear group \(\mathrm{PSL}(2,p)\). Then the problem has been reduced to considering the cases \((\mathrm{SL}(2,p),H)\) where \(H\) is a maximal subgroup of \(\mathrm{SL}(2,p)\). Since there is a one-one correspondence between the maximal subgroups of \(\mathrm{SL}(2,p)\) and those of \(\mathrm{PSL}(2,p)\), the finding has been further reduced to considering the maximal subgroups of \(\mathrm{PSL}(2,p)\) up to conjugacy. Next, the authors show that if \((\mathrm{SL}(2,p),H)\) is a strong Gelfand pair, then \(H\) must be a subgroup of \(\mathrm{U}(2,p)\). The proof is then completed after showing that if \(K\subseteq\mathrm{U}(2,p)\) is any proper subgroup and \(K\neq T\), then \((\mathrm{SL}(2,p), K)\) is not a strong Gelfand pair.