Irreducibility of tensor squares, symmetric squares and alternating squares.

*(English)*Zbl 1072.20013The authors investigate the question when the tensor square, the alternating square, or the symmetric square of an absolutely irreducible projective representation \(V\) of an almost simple group \(G\) is again irreducible. The information about such representations can be used in the study of the maximal subgroups of simple classical groups of Lie type.

Let \(R = R(l^f)\) be a finite classical group of Lie type and let \(G<R\) be a quasi-simple subgroup acting absolutely irreducibly on the natural module \(V\) of \(R\). Furthermore, write \(\widetilde\Lambda^2(V)\), \(\widetilde\Sigma^2(V)\) respectively \(\widetilde A(V)\) for the largest irreducible \(R\)-subquotient of \(\Lambda^2(V)\), \(\text{Sym}^2(V)\) respectively \(V\otimes V^*\). The authors study the cases when \(G\) acts irreducibly on \(V\) as well as on \(X(V)\), where \(X\in\{\widetilde\Lambda^2,\widetilde\Sigma^2,\widetilde A\}\).

It is known that the following families of examples do occur: (1) \(V\) is the heart of the natural permutation module of \(G=\text{Alt}_n\); (2) \(V\) is a Weil module of \(G=\text{Sp}_{2n}(q)\), \(q\in\{3,5,9\}\); (3) \(V\) is a Weil module of \(G=\text{SU}_n(q)\), \(q\in\{2,3\}\); (4) \(V\) is a module of dimension \((2^n-1)(2^{n-1}-1)/3\), \((2^n+1)(2^{n-1}+1)/3\) or \((2^{2n}-1)/3\) for \(G=\text{Sp}_{2n}(2)\) and \(X=\widetilde\Lambda^2\).

Now let \(l(G)\) denote the (known lower bounds for the) minimal dimensions of nontrivial representations. Then the main results can be formulated as follows:

Theorem 1.2. Let \(G\) be quasi-simple and \(V\) a nontrivial absolutely irreducible representation in characteristic \(l\geq 0\) (which is different from the defining characteristic if \(G\) is of Lie type). Let \(X=\widetilde A\) if \(V\) is not self-dual, and \(X\in\{\widetilde\Lambda^2,\widetilde\Sigma^2\}\) otherwise. Then one of the following holds: (a) \(X(V)\) is reducible; (b) \((G,V)\) are as in (1)-(4) above; (c) \(G=G(q)\) is classical, \(q\in\{2,4,8\}\), and \(\dim(V)\leq c\cdot l(G)^2\); (d) \(G=G(q)\) is exceptional, \(q\) is even, and \(\dim(V)\leq 4\cdot l(G)\); (e) \(G\) is on a known finite list of groups.

Theorem 1.3. Let \(G\) be a finite subgroup of a complex simple simply-connected Lie group \(\Gamma\) which is irreducible in all fundamental representations of \(\Gamma\). Assume that the dimension \(d\) of the natural module \(V\) for \(\Gamma\) is at least \(5\) and \(\Gamma\neq\text{Spin}_5(\mathbb{C})\cong\text{Sp}_4(\mathbb{C})\), \(\Gamma\neq\text{Spin}_6(\mathbb{C})\cong\text{SL}_4(\mathbb{C})\). Then up to a finite subgroup of the center \(Z(\Gamma)\) one of the following holds, where \(\overline G\) denotes the image of \(G\) in its action on the natural module for \(\Gamma\): (a) \(G=k^m.H\) is monomial on the natural module with \(k\geq 2\), \(d-1\leq m\leq d\), \(\Gamma=\text{SL}_d(\mathbb{C})\), and \(\text{Alt}_d\leq H\leq\text{Sym}_d\) or \(H\) as in Table 7.18 (a finite list of known groups). (b) \(\overline G=2^m.H\) is monomial on the natural module with \(d-1\leq m\leq d\), \(\Gamma=\text{Spin}_d(\mathbb{C})\), and \(\text{Alt}_d\leq H\leq\text{Sym}_d\) or \(H\) as in Table 7.18. (c) \(G=2^3.\text{SL}_3(2)\) and \(\Gamma=G_2(\mathbb{C})\). (d) \(G\leq 5_+^{1+2}:\text{SL}_2(5)\) and \(\Gamma=\text{SL}_5(\mathbb{C})\). (e) \(\overline G\leq 2_+^{1+6}.\text{Sym}_8\) and \(\Gamma=\text{Spin}_8(\mathbb{C})\). (f) \(G\) is almost quasi-simple and \((\Gamma,G)\) is on a known list (Table 7.22).

Let \(R = R(l^f)\) be a finite classical group of Lie type and let \(G<R\) be a quasi-simple subgroup acting absolutely irreducibly on the natural module \(V\) of \(R\). Furthermore, write \(\widetilde\Lambda^2(V)\), \(\widetilde\Sigma^2(V)\) respectively \(\widetilde A(V)\) for the largest irreducible \(R\)-subquotient of \(\Lambda^2(V)\), \(\text{Sym}^2(V)\) respectively \(V\otimes V^*\). The authors study the cases when \(G\) acts irreducibly on \(V\) as well as on \(X(V)\), where \(X\in\{\widetilde\Lambda^2,\widetilde\Sigma^2,\widetilde A\}\).

It is known that the following families of examples do occur: (1) \(V\) is the heart of the natural permutation module of \(G=\text{Alt}_n\); (2) \(V\) is a Weil module of \(G=\text{Sp}_{2n}(q)\), \(q\in\{3,5,9\}\); (3) \(V\) is a Weil module of \(G=\text{SU}_n(q)\), \(q\in\{2,3\}\); (4) \(V\) is a module of dimension \((2^n-1)(2^{n-1}-1)/3\), \((2^n+1)(2^{n-1}+1)/3\) or \((2^{2n}-1)/3\) for \(G=\text{Sp}_{2n}(2)\) and \(X=\widetilde\Lambda^2\).

Now let \(l(G)\) denote the (known lower bounds for the) minimal dimensions of nontrivial representations. Then the main results can be formulated as follows:

Theorem 1.2. Let \(G\) be quasi-simple and \(V\) a nontrivial absolutely irreducible representation in characteristic \(l\geq 0\) (which is different from the defining characteristic if \(G\) is of Lie type). Let \(X=\widetilde A\) if \(V\) is not self-dual, and \(X\in\{\widetilde\Lambda^2,\widetilde\Sigma^2\}\) otherwise. Then one of the following holds: (a) \(X(V)\) is reducible; (b) \((G,V)\) are as in (1)-(4) above; (c) \(G=G(q)\) is classical, \(q\in\{2,4,8\}\), and \(\dim(V)\leq c\cdot l(G)^2\); (d) \(G=G(q)\) is exceptional, \(q\) is even, and \(\dim(V)\leq 4\cdot l(G)\); (e) \(G\) is on a known finite list of groups.

Theorem 1.3. Let \(G\) be a finite subgroup of a complex simple simply-connected Lie group \(\Gamma\) which is irreducible in all fundamental representations of \(\Gamma\). Assume that the dimension \(d\) of the natural module \(V\) for \(\Gamma\) is at least \(5\) and \(\Gamma\neq\text{Spin}_5(\mathbb{C})\cong\text{Sp}_4(\mathbb{C})\), \(\Gamma\neq\text{Spin}_6(\mathbb{C})\cong\text{SL}_4(\mathbb{C})\). Then up to a finite subgroup of the center \(Z(\Gamma)\) one of the following holds, where \(\overline G\) denotes the image of \(G\) in its action on the natural module for \(\Gamma\): (a) \(G=k^m.H\) is monomial on the natural module with \(k\geq 2\), \(d-1\leq m\leq d\), \(\Gamma=\text{SL}_d(\mathbb{C})\), and \(\text{Alt}_d\leq H\leq\text{Sym}_d\) or \(H\) as in Table 7.18 (a finite list of known groups). (b) \(\overline G=2^m.H\) is monomial on the natural module with \(d-1\leq m\leq d\), \(\Gamma=\text{Spin}_d(\mathbb{C})\), and \(\text{Alt}_d\leq H\leq\text{Sym}_d\) or \(H\) as in Table 7.18. (c) \(G=2^3.\text{SL}_3(2)\) and \(\Gamma=G_2(\mathbb{C})\). (d) \(G\leq 5_+^{1+2}:\text{SL}_2(5)\) and \(\Gamma=\text{SL}_5(\mathbb{C})\). (e) \(\overline G\leq 2_+^{1+6}.\text{Sym}_8\) and \(\Gamma=\text{Spin}_8(\mathbb{C})\). (f) \(G\) is almost quasi-simple and \((\Gamma,G)\) is on a known list (Table 7.22).

Reviewer: Wolfgang Lempken (Essen)

##### MSC:

20C33 | Representations of finite groups of Lie type |

20C20 | Modular representations and characters |