Inclusion matrix of \(k\) vs. \(l\) affine subspaces and a permutation module of the general affine group
The author studies some possibilities of computing the rank of inclusion matrices of affine subspaces of a finite dimensional vector space over a finite field \(\mathbb{F}_ q\). The main results are: Theorem 3.1. Let \(\mathbb{F}_ q\) be a field with \(q\) elements, \(q=p^ d\), \(p\) being a prime. Let \(0 \leq k\), \(l<n\). \(A^{k,l}_ n\) be the inclusion matrix of \(k\)- vs. \(l\)-linear subspaces of \(\mathbb{F}_ q^{(n)}\) and \(B_ n^{k,l}\) be the inclusion matrix of \(k\)- vs. \(l\)-affine subspaces of \(\mathbb{F}_ q^{(n)}\). Then \[ \text{rank}_ K(B_ n^{k,l})=\text{rank}_ K(A_ n^{k,l})+(q^ n-1)\text{rank}_ K (A_{n-l}^{k,1}). \] Theorem 3.2. With the same hypotheses, the following hold: (i) If \(k\leq l\), \(k+l<n\), then \[ \text{rank}_ K(B_ n^{k,l})=\sum_{i\in Y_{k,l}} \left({n \brack i}_ qq^{n-i}- {n\brack i-1}_ qq^{n-(i-1)}\right), \] where \(Y_{k,l}=\left\{i:0 \leq i \leq k,{l-i \brack k-i}_ q\neq 0\right.\) in the field \(\left. K \right\}\). (ii) If \(k\leq l\), \(k+l\geq n\), then \[ \text{rank}_ K(B_ n^{k,l})=\sum_{j \in X^ n_{k,l}} \left({n \brack j}_ qq^{n- j}-{n \brack j+1}_ qq^{n-(j+1)} \right), \] where \(X^ n_{k,l}=\left\{j:l \leq j \leq n,{j-k \brack j-l}_ q \neq 0\right.\) in the field \(\left. K \right\}\).
scientific article; zbMATH DE number 217950