Schubert varieties, linear codes and enumerative combinatorics

Schubert varieties, linear codes and enumerative combinatorics

We consider linear error correcting codes associated to higher-dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have...

Linear codes from Schubert Varieties

A family of linear \Algebraic{Geometric" codes is constructed from line bundles on Schubert varieties and analyzed using techniques from Representation Theory. AMS(MOS) 1991 Subject Classi cation (Primary) 94B27 (Secondary) 14M99, 20G10. Research partially supported by the US Air Force O ce of Scienti c Research. Research partially supported by the Isaac Newton Institute for Mathematical Scienc...

Combinatorics in Schubert varieties and Specht modules

This thesis consists of two parts. Both parts are devoted to finding links between geometric/algebraic objects and combinatorial objects. In the first part of the thesis, we link Schubert varieties in the full flag variety with hyperplane arrangements. Schubert varieties are parameterized by elements of the Weyl group. For each element of the Weyl group, we construct certain hyperplane arrangem...

Combinatorics Enumerative Combinatorics 05

Enumerative and Algebraic Combinatorics

Enumeration, alias counting, is the oldest mathematical subject, while Algebraic Combinatorics is one of the youngest. Some cynics claim that Algebraic Combinatorics is not really a new subject but just a new name given to Enumerative Combinatorics in order to enhance its (former) poor image, but Algebraic Combinatorics is in fact the synthesis of two opposing trends: abstraction of the concret...