Degenerations in a class of matrix varieties and prinjective modules
Let \(K\) be an algebraically closed field and \(I\equiv(I,\preceq)\) a finite partially ordered set. Assume that \(\max I=\{p_1,\dots,p_t\}\) is the set of all maximal elements of \(I\) and \(I\setminus\max I=\{1,\dots,n\}\). Given a vector \(v=(v_1,\dots,v_n,v_{p_1},\dots,v_{p_t})\in\mathbb{N}^I\), denote by \({\mathbf{Mat}}^I_v\) the irreducible affine \(K\)-variety consisting of all bipartitioned matrices of the form \[ A=\begin{bmatrix} A_{1p_1}&A_{2p_1}&\cdots&A_{np_1}\\ \vdots &\vdots&\ddots&\vdots\\ A_{1p_t}&A_{2p_t}&\cdots&A_{np_t}\end{bmatrix} \] with coefficients in \(K\), where \(A_{ip_r}\) is the matrix with \(v_i\) rows, \(v_{p_r}\) columns such that \(A_{ip_r}=0\) if \(i\not\preceq p_r\). Let \({\mathbf G}^I_v\) be the parabolic algebraic group generated by the following \(I\)-elementary transformations on matrices in \({\mathbf {Mat}}^I_v\). \((E_1)\) All elementary transformations on rows inside each horizontal block of the matrix \(A\). \((E_2)\) All elementary transformations on columns from the \(i\)-th block to the \(j\)-th block, for all \(i\preceq j\in I\setminus\max I\). Finally, let \(*\colon{\mathbf G}^I_v\times{\mathbf{Mat}}^I_v\to{\mathbf{Mat}}^I_v\) be the obvious algebraic group action. One of the main aims of the paper is to give a simple description of degenerations of \({\mathbf G}^I_v\) orbits in the variety \({\mathbf{Mat}}^I_v\). This is done in terms of homomorphisms and extensions of prinjective \(KI\)-modules of finite dimension, where \(KI\) is the incidence \(K\)-algebra of \(I\). Here, by a prinjective \(KI\)-module we mean a right \(KI\)-module \(X\) of projective dimension at most one such that there is a minimal projective presentation \(0\to P_1\to P_0\to X\to 0\), where \(P_1\), \(P_0\) are projective and \(P_1\) is a direct sum of simple modules. A complete solution is given in case the integral Tits quadratic form \(q_I\colon\mathbb{Z}^I\to\mathbb{Z}\) defined by the formula \[ q_I(x)=\sum _{i\in I}x_i^2+\sum_{i\prec j\not\in\max I}x_ix_j-\sum_{p \in\max I}(\sum_{i\prec p}x_i)x_p \] is weakly positive or \(I\) is a minimal poset such that \(q_I(v)=0\), for some non-zero vector \(v\in\mathbb{N}^I\). We recall that \(q_I\) is said to be weakly positive if \(q_I(w)>0\) for any non-zero vector \(w\in\mathbb{N}^I\).
scientific article; zbMATH DE number 1930292
scientific article