Symplectic polynomial invariants of one or two matrices of small size (Q2923070)

Let \(\text{Sp}_{2n}\) be the symplectic group of \(2n\times 2n\) complex matrices acting by simultaneous conjugation on \(k\)-tuples of \(2n\times 2n\) matrices. A classical result [\textit{C. Procesi}, Adv. Math. 19, 306--381 (1976; Zbl 0331.15021)] gives that the algebra \(\bar C\) of \(\text{Sp}_{2n}\)-invariants under the above action of \(\text{Sp}_{2n}\) is generated by traces of products of generic \(2n\times 2n\) matrices \(x\) and their images \(x^{\ast}\) under the symplectic involution, with explicit upper bounds on the degree of the products. Nevertheless, explicit sets of generators and other characteristics of \(\bar C\) are known only in a handful of low-dimensional cases. The vector space \(V={\mathfrak g}\) of all \(2n\times 2n\) matrices is a direct sum of two \(\text{Sp}_{2n}\)-invariant subspaces, \({\mathfrak g}={\mathfrak k}\oplus{\mathfrak p}\), where \(\mathfrak k\) and \(\mathfrak p\) consist, respectively, of all skew-symmetric and symmetric matrices with respect to the symplectic involution. In the paper under review the author considers the action of \(\text{Sp}_{2n}\) by simultaneous conjugation on \(k_1\) copies of \(\mathfrak p\) and \(k_2\)-copies of \(\mathfrak k\). Then the algebra of \(\text{Sp}_{2n}\)-invariants has a natural \({\mathbb Z}^{k_1+k_2}\)-multigrading. Using the classical Molien-Weyl formula, he computes the Poincaré (or Hilbert) series of the algebra of invariants (as a multigraded or simply graded vector space) for \(n=2\) and \(k_1+k_2=3\). (It follows from \textit{G. W. Schwarz} [Invent. Math. 49, 167--191 (1978; Zbl 0391.20032)] that when \(k_2=0\) and \(k_1\leq 5\) the algebra of invariants is isomorphic to the polynomial algebra; \textit{B. Broer} [J. Algebra 168, No. 1, 43--70 (1994; Zbl 0817.14028)] handled the case \(k_1=0\), \(k_2=2\), and the other cases with \(k_1+k_2=2\) were completed by \textit{A. Berele} and \textit{R. M. Adin} [Isr. J. Math. 134, 93--125 (2003; Zbl 1042.16011)]). Then the author constructs explicit minimal sets of generators of the algebra of invariants for \(n=k=2\) and for \(n=3\), \(k=1\). (Before minimal systems of generators were known only for \(n=1\), when \(\text{Sp}_2=\text{SL}_2\) and for \(n=2\), \(k=1\).) For \(n=k(=k_1=k_2)=2\) the minimal system of generators consists of 136 traces of products of degree \(\leq 9\) and for \(n=3\), \(k(=k_1=k_2)=1\) the system consists of 28 traces. For \(n=3\), \(k=1\) the author constructs a homogeneous system of 15 parameters (which generate the polynomial algebra \(\bar C^{\#}\) in 15 variables) and shows that \(\bar C\) is a free \(\bar C^{\#}\)-module of rank 36 (with explicit generators again). This is an example of a Hironaka decomposition of \(\bar C\) with respect to \(\bar C^{\#}\). Finally, the author finds the only defining relation of minimal degree (of bidigree (6,8)) among these generators. The computations have been performed using MAPLE. They are based on simple ideas and are well presented in the exposition. It is clear that the methods work also for algebras of matrix invariants for other classical groups.