Linear preserver problems and algebraic groups
Let \(M_n=M_n(K)\) be the space of \(n\) by \(n\) matrices over an algebraically closed field \(K\) of characteristic 0. We have a direct decomposition \(M_n=M^0_n\oplus K\cdot 1\), where 1 denotes the identity matrix, and \(M^0_n\) the subspace defined by \(\text{tr}(x)=0\). We denote by \(\text{GL}(n^2)\) the group of vector space automorphisms of \(M_n\) and by \(\text{GL}(n^2-1)\) its subgroup which fixes 1 and preserves \(M^0_n\). By \(G\) we denote the subgroup consisting of all similarity transformations \(x\mapsto axa^{-1}\) where \(a\in\text{GL}(n)\). Note that \(G\cong\text{PSL}(n)\). We define \(\tau\in\text{GL}(n^2-1)\) to be the transposition map and we write \(x'\) for \(\tau(x)\). We classify all algebraic subgroups \(\Gamma\) of \(\text{GL}(n^2)\) containing \(G\). Let \(O(n^2)\) be the subgroup of \(\text{GL}(n^2)\) which preserves the non-degenerate symmetric bilinear form \(\text{tr}(xy) \), \(x, y\in M_n\), \(O(n^2-1)\) its subgroup consisting of operators which fix the identity matrix 1. Let \(P\) (resp. \(Q\)) be the subgroup of \(\text{GL}(n^2)\) which acts trivially on \(M^0_n\) and \(M_n/M^0_n\) (resp. \(K\cdot 1\) and \(M_n/K\cdot 1\)). By \(T\) we denote the 2-dimensional subtorus of \(\text{GL}(n^2)\) which acts by scalar transformations on \(M^0_n\) and \(K \cdot 1\), and we set \(T_1=T\cap\text{SL}(n^2)\). If \(n=4\), there is an overgroup \(\Lambda\cong\text{SL}(6)/\langle -1\rangle\) of \(G\) in \(\text{SL}(15)\). Theorem A. Let \(\Gamma\) be a proper connected algebraic subgroup of \(\text{SL}(n^2)\) containing \(G=\text{PSL}(n)\). Then (i) if reducible, \(\Gamma\) is one of the groups \(H\), \(HP\), \(HQ\), \(HT_1\), \(HPT_1\), \(HQT_1\), where \(H=G\), \(\text{SO}(n^2-1)\), \(\text{SL}(n^2-1)\), or \(\Lambda\) if \(n=4\); (ii) if irreducible, \(\Gamma\) is one of the groups \(t\text{SO}(n^2)t^{-1}\), \(t(\text{SL}(n)\otimes\text{SL}(n)^{\text{opp}})t^{-1}\), \(t\in T_1\). For applications to linear preserver problems it is necessary to have a description of all, not only connected, algebraic subgroups \(\Gamma\subset\text{GL}(n^2)\) containing \(G\). This problem reduces essentially to the computation of normalizers in \(\text{GL}(n^2)\) of the various groups listed in Theorem A. All these normalizers are explicitly determined (Theorem B). The above mentioned results are applied to obtain solutions of several new linear preserver problems over arbitrary fields of characteristic 0. It is also shown that solutions of several classical linear preserver problems follow immediately from Theorems A and B. [Concerning linear preserver problems see the contributions by various authors in Linear Multilinear Algebra 33, No. 1/2 (1992; Zbl 0766.15014; Zbl 0766.15015; Zbl 0767.15006; Zbl 0767.15008; Zbl 0767.15009; Zbl 0767.15010; Zbl 0768.15020 and Zbl 0784.15007)]. Several typical results are given below, where \(F\) denotes an arbitrary field of characteristic 0. Theorem 3. Let \(X_F \subset M_n (F)\) be the set of regular elements and \(\sigma\) a nonsingular linear transformation of \(M_n(F)\). If \(n > 2\), \(\sigma\) preserves \(X_F\) iff \(\sigma\) has the form (1) \(\sigma (x)=\alpha axa^{-1}+f(x) \cdot 1\), or (2) \(\sigma (x)=\alpha ax' a^{-1}+f(x) \cdot 1\), where \(\alpha \in F^*\), \(a \in \text{GL} (n,F)\), and \(f\) is a linear function on \(M_n(F)\) such that \(f(1) \neq -\alpha\). If \(n=2\), then \(\sigma\) preserves \(X_F\) iff \(\sigma\) leaves the subspace \(F \cdot 1\) invariant. Theorem 4. Let \((X_r)_F \subset M_n (F)\), \(r < n\), be the set of matrices having at most \(r\) distinct eigenvalues in some algebraic closure \(K\) of \(F\). Let \(\sigma\) be a nonsingular linear transformation of \(M_n (F)\). Then \(\sigma ((X_r)_F) \subset (X_r)_F\) iff \(\sigma\) has the form (1) or (2). Theorem 8. Let \(X_F\) be the set of elements of finite order in \(\text{GL} (n, F) \subset M_n(F)\). If \(\sigma\) is a nonsingular linear transformation of \(M_n (F)\) and \(\sigma (X_F) \subset X_F\), then there exists \(a \in \text{GL}_n (F)\) and a root of unity \(\alpha \in F^*\) such that either \(\sigma(x)=\alpha axa^{-1}\) or \(\sigma (x)=\alpha a x' a^{-1}\) for all \(x\). A first example of a linear preserver problem in \(M_4\) is given for which the group \(\Lambda \cong \text{SL} (6) / \langle - 1\rangle\) appears as the main component of the stabilizer group.
scientific article; zbMATH DE number 840180
scientific article