Sudhir R. Ghorpade

Higher Weights of Grassmann Codes

Using a combinatorial approach to studying the hyperplane sections of Grassmannians, we give two new proofs of a result of Nogin concerning the higher weights of Grassmann codes. As a consequence, we obtain a bound on the number of higher dimensional subcodes of the Grassmann code having the minimum Hamming norm. We also discuss a generalization of Grassmann codes.

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

We consider the question of determining the maximum number of points on sections of Grassmannians over finite fields by linear subvarieties of the Plucker projective space of a fixed codimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. A basic tool used is a characterization of decomposable subspaces of exterior powers, that is, subspaces in which every nonzero element is decomposable. Also, we use a generalization of the Griesmer-Wei bound that is proved here for arbitrary linear codes.

Hilbert functions of points on Schubert varieties in the symplectic Grassmannian

We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian.

Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields

Abstract We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed.

The Hilbert Series of Pfaffian Rings

We give three determinantal expressions for the Hilbert series as well as the Hilbert function of a Pfaffian ring, and a closed form product formula for its multiplicity. An appendix outlining some basic facts about degeneracy loci and applications to multiplicity formulae for Pfaffian rings is also included.

Number of solutions of equations over finite fields and a conjecture of Lang and Weil

A brief survey of the conjectures of Weil and some classical estimates for the number of points of varieties over finite fields is given. The case of partial flag manifolds is discussed in some details by way of an example. This is followed by a motivated account of some recent results on counting the number of points of varieties over finite fields, and a related conjecture of Lang and Weil. Explicit combinatorial formulae for the Betti numbers and the Euler characteristics of smooth complete intersections are also discussed.

Automorphism groups of Grassmann codes

Abstract We use a theorem of Chow (1949) on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question of Beelen et al. (2010) concerning the permutation automorphism groups of affine Grassmann codes. Finally, we prove an analogue of Chowʼs theorem for the case of Schubert divisors in Grassmannians and then use it to determine the automorphism group of linear codes associated to such Schubert divisors. In the course of this work, we also give an alternative short proof of MacWilliams theorem concerning the equivalence of linear codes and a characterization of maximal linear subspaces of Schubert divisors in Grassmannians.

Young bitableaux, lattice paths and Hilbert functions

Abstract A recent result on the enumeration of p -tuples of nonintersecting lattice paths in an integral rectangle is used to deduce a formula of Abhyankar for standard Young bitableaux of certain type, which gives the Hilbert function of a class of determinantal ideals. The lattice path formula is also shown to yield the numerator of the Hilbert series of these determinantal ideals and the h -vectors of the associated simplicial complexes. As a consequence, the a -invariant of these determinantal ideals is obtained in some cases, extending an earlier result of Grabe. Some problems concerning generalizations of these results to ‘higher dimensions’ are also discussed. In an appendix, the equivalence of Abhyankars formula for unitableaux of a given shape and a formula of Hodge, obtained in connection with his determination of Hilbert functions of Schubert varieties in Grassmannians, is outlined.

Abhyankar’s Work on Young Tableaux and Some Recent Developments

This paper is meant to be a survey of Abhyankar’s work on Young tableaux and some of the subsequent developments motivated by it. Leaving the technical details to the remaining sections, we now attempt to give an overview by narrating some background, a little bit of history, and some stories in the two paragraphs below. In keeping with the spirit of Abhyankar’s work on Young tableaux, much of this paper is of an elementary nature except perhaps for a few words thrown here and there.

Duals of Affine Grassmann Codes and Their Relatives

Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work by Beelen Here, we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results by Beelen concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes.