Poincaré series of semi-invariants of \(2\times 2\) matrices
Let \(\text{SL}_2(\mathbb{C})\times\text{SL}_2(\mathbb{C})\) act on \(m\)-tuples of \(2\times 2\) matrices. The ring of invariants can be seen as the ring of semi-invariants for the Kronecker quiver with \(m\) arrows (i.e., 2 vertices and \(m\) arrows from one vertex to the other) and dimension vector \((2,2)\). The author studies the Poincaré series of the invariant ring. He uses Weyl's integration formula and iterated application of the Residue Theorem to find a nice formula for the Poincaré series as the quotient of two determinantal expressions. Generators for the invariant ring are described. In general generators of rings of semi-invariants for quiver representations were described by \textit{M. Domokos} and \textit{A. N. Zubkov} [Transform. Groups 6, No.~1, 9-24 (2000; Zbl 0984.16023)], \textit{A. Schofield} and \textit{M. Van den Bergh} [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; Zbl 1004.16012)] and \textit{H. Derksen} and \textit{J. Weyman} [J. Am. Math. Soc.~13, No.~3, 467-479 (2000; Zbl 0993.16011)]. There is a natural action of \(\text{GL}_m(\mathbb{C})\) on the set of \(m\)-tuples of \(2\times 2\) matrices. So the invariant ring is a \(\text{GL}_m(\mathbb{C})\)-module, and the Poincaré series can be viewed as a sum over Schur functions. This is applied to get results for \(\text{SL}_2(\mathbb{C})\times\text{SL}_2(\mathbb{C})\times\text{SL}_m(\mathbb{C})\)-invariants of the representation \(\mathbb{C}^2\times\mathbb{C}^2\times\mathbb{C}^m\).
scientific article; zbMATH DE number 1453261
scientific article