Schubert unions in Grassmann varieties

Abstract

We study subsets of Grassmann varieties G(l, m) over a field F , such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study the linear spans of, and in case of positive characteristic, the number of Fq-rational points on such unions. Moreover we study a geometric duality of such unions, and give a combinatorial interpretation of this duality. We discuss the maximum number of Fq-rational points for Schubert unions of a given spanning dimension, and we give some applications to coding theory. We define Schubert union codes, and study the parameters and support weights of these codes and of the well-known Grassmann codes.1991 Mathematics Subject Classification. 14M15 (05E15, 94B27).

Decomposable subspaces, linear sections of Grassmann varieties, and higher weights of Grassmann codes

“…To this end, we provide, toward the end of this paper, an initial tangible goal by stating conjectural formulae for d r (C(2, m)) when μ + 1 r 2μ − 3, and also when k − 2μ + 3 r k − μ − 1. It may be noted that these conjectural formulae, and of course both (5) and (6), corroborate a conjecture of Hansen, Johnsen and Ranestad [7,8] that in most cases, the differences of consecutive higher weights of Grassmann codes is a power of q.…”

“…In general, we believe that, as predicted in most cases by Hansen, Johnsen and Ranestad [7,8], the difference d r (C( , m)) − d r−1 (C( , m)) is always a power of q, and that determining d r (C( , m)) from d r−1 (C( , m)) is a matter of deciphering the correct term of the Gaussian binomial coefficient (which can be written as a finite sum of powers of q) that gets added to d r−1 (C( , m)). …”

“…Recently, Hansen, Johnsen and Ranestad [7,8] have observed that a dual result holds as well, namely,…”

“…For example, if = 2 and we assume (without loss of generality) that m 4, then μ = m − 1, and d r (C( , m)) for m r < m−1 2 are not known, except that in the first nontrivial case, Hansen, Johnsen and Ranestad [7] have shown by clever algebraic-geometric arguments that…”

On Lagrangian–Grassmannian codes

“…Schubert codes, a generalization of the Grassmann codes, were introduced by Ghorpade and Lachaud in 1998 as codes associated to projective systems defined by Schubert varieties, and they give an upper bound for its minimum distance with a conjecture for its actual value [4,5]. See also [7][8][9][10][11]17] for recent developments, where we would like to call the attention to the problem of the determination of the complete weight hierarchy of linear codes associated to these varieties, and for the study of unions of Schubert cycles in Grassmann varieties.…”