Wronskians, cyclic group actions, and Ribbon tableaux (Q2839387)

The paper under review is about the intriguing combinatorics relating the enumerations of fixed points in the fibers of the \textit{Wronski map} \(\text{Wr}: \mathrm{Gr}(d, n)\rightarrow {\mathbb P}^{d(n-d)}\), acted on by a finite subgroup \(C_r\) of \(\mathrm{PGL}_2({\mathbb C})\), with the enumeration of \textit{standard ribbon tableaux}. The main result consists in computing the number of \(C_r\)-fixed points in the fiber of the Wronski map through the number of standard \(r\)-ribbon tableaux of rectangular shape \((n-d)^d\). If \((n-d)d=rs\), it is also shown that such a number can be seen as the number of standard Young tableaux of the same rectangular shape which are left invariant by \(s\) iteration of the \textsl{jeu de taquin} (see e.g. [Adv. Math. 224, No. 3, 827--862 (2010; Zbl 1218.14048)]. The rest of this review will be devoted to supply a few vocabulary in order to let the non specialist reader guessing the flavor of the results.NEWLINENEWLINECombinatorics first. \textsl{Standard ribbon tableaux} are generalizations of \textsl{Young tableaux} (see [\textit{W. Fulton}, Young tableaux. London Mathematical Society Student Texts. 35. Cambridge: Cambridge University Press. (1997; Zbl 0878.14034)]). If \(\lambda:=(\lambda_1\geq\lambda_2\dots\geq \lambda_p)\) is a partition of the integer \(|\lambda|=\lambda_1+\dots+\lambda_p\), its \textsl{Young diagram} \(Y(\lambda)\) is an array of left justified boxes, the first row with \(\lambda_1\) boxes, the second one with \(\lambda_2,\dots\), the \(p\)th one with \(p\) boxes. Partitions are partially ordered by declaring that \(\lambda\supset \mu\) if the Young diagram of \(\lambda\) contains that of \(\mu\). The notation \(Y(\lambda/ \mu)\) means the diagram resulting by deleting from \(Y(\lambda)\) the boxes of \(Y(\mu)\) and \(|Y(\lambda/\mu)|\) means the number of boxes of \(Y(\lambda/\mu)\). A tableau of shape \(\lambda/\mu\) is a filling of \(Y(\lambda/\mu)\) with positive integer entries that are weakly increasing along rows and down columns. It is said to be \textsl{standard} if its entries are exactly the numbers \(1,\dots, |Y(\lambda/\mu)|\). Suppose now that \(|\lambda\setminus\mu|=rl\). A standard \(r\)-\textsl{ribbon tableau} is a tableau of shape \(\lambda\setminus\mu\) such that i) the entries are \(1,2,\dots,l\) and ii) the shape determined by the entries labelled \(i\) is a (connected) \(r\)-ribbon, i.e. a connected skew shape with \(r\) boxes that does not contain a \(2\times 2\) square. When \(r=1\), the definition coincides with the definition of a standard Young tableaux.NEWLINENEWLINEOn the other more geometrical side, let \(n\) be a positive integer and \(V:={\mathbb C}_{n-1}[z]\) be the complex vector space of polynomials of degree smaller or equal than \(n\). For \(1\leq d\leq n\), let \( \mathrm{Gr}(d, {\mathbb C}_{n-1}[z])\) denote the complex Grassmannian of \(d\)-dimensional vector subspaces of \(V\). If \(x\in \mathrm{Gr}(d, {\mathbb C}_{n-1}[z])\) and \((v_1,\dots, v_d)\) is a basis of \(x\), the Wronskian determinant NEWLINE\[NEWLINE \left|\begin{matrix} v_1&\dots&v_d\cr v'_1&\dots&v'_d\cr \vdots&\ddots&\vdots\cr v^{d-1}_1&\dots&v_d^{(d-1)}\end{matrix}\right|, NEWLINE\]NEWLINE is a polynomial of degree \(N:=d(n-d)\). The derivatives are taken with respect to the indeterminate \(z\). Changing the basis of \(x\), the Wronskian gets multiplied by a non zero complex number. Thus \(x\in \mathrm{Gr}(d, {\mathbb C}_{n-1}[z])\) determines a well defined point \(\text{Wr}(x)\in {\mathbb P}^N\), said to be the Wronskian of \(x\). The Wronski map is the map \(\text{Wr}: \mathrm{Gr}(d, {\mathbb C}_{n-1}[z])\rightarrow {\mathbb P}^N\) given by \(x \mapsto \text{Wr}(x)\). It is a finite morphism of degree \(d(n-d)\). It is easily seen that the Wronski map is equivariant with respect to a natural action of the group \(\mathrm{PGL}_2({\mathbb C})\) on both \(\mathrm{Gr}(d, {\mathbb C}_{n-1}[z])\) and \({\mathbb P}^N\). For \(r>1\), let \(C_r\) be a cyclic subgroup of order \(r\) of \(\mathrm{PGL}_2({\mathbb C})\). As announced at the beginning of the review, the paper is concerned with the number of \(C_r\)-fixed points in the fiber of \(\text{Wr}\) over a \(C_r\)-fixed point of \({\mathbb P}^N\). When \(r=2\), the computation gives the number of real points in the fibre of the Wronski map over a real polynomial with purely imaginary roots. This extremely interesting paper is also related with the best literature produced in combinatorics and in the geometry of the Wronski maps, as it is clearly shown in the reference list, starting from the very important work by \textit{A. Eremenko} and \textit{A. Gabrielov} on the subject [Discrete Comput. Geom. 28, No. 3, 331--347 (2002; Zbl 1004.14011)]. The restriction of the Wronski map to Richardson varieties, whose definition will not be recalled here due to the limited purposes of a review, is also considered. The paper is ideally divided into two part. The part regarding the restriction of the Wronski map to the \(C_r\) fixed points in the fibers over \(C_r\)-fixed points, for \(r>2\), is in Sections 4, 5, 6, while sections 2 and 3 are especially concerned with involutions-fixed points.