Bulk irreducibles of the modular group (Q2788751)

Let \(\Gamma\) be the modular group, i.e., \(\Gamma =\mathrm{PSL}_{2}(\mathbb{Z})\). Let \(\mathrm{rep}_{n}\Gamma\) be the \(n\)-dimensional representations of \(\Gamma\), and let \(\mathrm{iss}_{n}\Gamma\) be the affine GIT\ quotient \(\mathrm{iss} _{n}\Gamma =\mathrm{rep}_{n}G/PGL_{n}\). Then \(\mathrm{iss}_{n}\Gamma\) describes the isomorphism classes of semisimple \(\Gamma\)-representations of dimension \(n\), and \(\mathrm{iss}_{n}\Gamma\) decomposes into a (disjoint) union of irreducible components \(\mathrm{iss}_{\alpha}\Gamma\), where \(\alpha =(a,b;x,y,z)\), \(a+b=n=x+y+z\) is a dimension vector. With this notation, if \(xyz\neq 0\) and \(\max(x,y,z)\leq \min(a,b)\) then \(\mathrm{iss}_{\alpha}\Gamma\) contains an open subset of simple representations and has dimension \(1+n^{2}-(a^{2}+b^{2}+x^{2}+y^{2}+z^{2})\), a quantity denoted \(d_{\alpha}\).NEWLINENEWLINEThe principal result of this paper is the construction, for each component \(\mathrm{iss}_{\alpha}\Gamma\) of \(\mathrm{iss}_{n}\Gamma\) which contains irreducible representations, of an é tale rational map \(\mathbb{A}^{d_{\alpha}}\dashrightarrow \mathrm{iss}_{\alpha}\Gamma\) whose image contains a Zariski open dense subset of \(\mathrm{iss}_{\alpha}\Gamma\). The map is made explicit, however the definition is different for the cases \(a=b\) and \(a\neq b\). This result extends works of \textit{I. Tuba} and \textit{H. Wenzl} [Pac. J. Math. 197, No. 2, 491--510 (2001; Zbl 1056.20025)], who proved it for \(n\leq 5\), and \textit{L. Le Bruyn} [J. Pure Appl. Algebra 215, No. 5, 1003--1014 (2011; Zbl 1260.20058)] for \(n\leq 11\). This provides insight as well into the the irreducible representations of the \(3\)-string braid group \(B_{3}\) as their classification reduces to the classification of irreducible \(\Gamma\)-representations.