Strong Gelfand pairs of \(\mathrm{SL} (2, p)\). (English) Zbl 1517.20070
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 \(p>11\),
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 \(p>11\),
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- if \(p\equiv 1\pmod{4}\), then the only strong Gelfand pair is given by \(\left(\mathrm{SL}(2,p),\mathrm{U}(2,p)\right)\),
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- 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 \(\left(\mathrm{SL}(2,p),T\right)\).
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- 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 \((\mathrm{SL}(2,2), C_3)\).
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- For \(p=3\), the strong Gelfand pairs are \((\mathrm{SL}(2,3),C_3)\), \((\mathrm{SL}(2,3),C_6)\) and \((\mathrm{SL}(2,3),\mathrm{U}(2,3))\).
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- For \(p=5\), the strong Gelfand pairs are \((\mathrm{SL}(2,5),\mathrm{SL}(2,3))\) and \((\mathrm{SL}(2,5),\mathrm{U}(2,5))\).
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- 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.
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- 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 \(2I=\langle r,s,t|r^2=s^3=t^5=rst \rangle\).
Reviewer: Saikat Panja (Prayāgrāj)
MSC:
20G05 | Representation theory for linear algebraic groups |
20G40 | Linear algebraic groups over finite fields |
20C15 | Ordinary representations and characters |
Keywords:
strong Gelfand pair; special linear group; upper triangular group; characters; finite groupsSoftware:
MagmaReferences:
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