The expected weighted dimension of a sum of vector spaces (Q2785934)

Let \(W\) be an \(n\)-dimensional vector space, for some positive integer \(n\), over a finite field \(F\) with \(q\) elements. It was shown by \textit{D. E. Dobbs} and \textit{M. J. Lancaster} [(*) Bull. Aust. Math. Soc. 45, No. 3, 467-478 (1992; Zbl 0743.15003), Theorem 3.4] that if each subspace of \(W\) is deemed equally likely, then the expected dimension of a subspace of \(W\) is \(n/2\).NEWLINENEWLINENEWLINEOne may ask whether such a result persists if the underlying probability function is altered so as to account for the varying ``sizes'' of the subspaces of \(W\). More precisely, let \(\nu_i\) denote the number of \(i\)-dimensional subspaces of \(W\) (for \(i=0,1, \dots,n)\), let \(\{w_i\}\) be a set of suitable ``weights,'' and consider the probability function \(p\) given by \(p(j)= w_j \nu_j/ \sum^n_{i=0} w_i\nu_i\) for \(j=0,1, \dots, n\). If \(w_i=i\) (resp., \(q^i)\), we say that the probability has been weighted by dimension (resp., weighted by cardinality). With the above notation, the expected dimension of a subspace of \(W\) is \(\sum^n_{j=0} jp(j)\). In contrast with (*), where we assumed that \(w_i=1\) for all \(i\), this expression may depend on \(q\) as well as on \(n\).NEWLINENEWLINENEWLINEAccordingly one may ask for the limiting value of this expectation as \(q\to \infty\). It is shown that this limit is \((n+1)/2\) if one weights by cardinality; but this limit depends on the parity of \(n\) if one weights by dimension, producing the limit \(n/2\) if \(n\) is even and \(n/2 +1/2n\) if \(n\) is odd. Moreover, we show that with suitable weighting, this limit may take on any value between 0 and \(n\) and, in fact, may not exist if \(n>1\).NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].