By Naihuan Jing

Algebraic combinatorics has advanced into probably the most energetic parts of arithmetic over the last numerous a long time. Its fresh advancements became extra interactive with not just its conventional box illustration conception but in addition algebraic geometry, harmonic research and mathematical physics.

This publication provides articles from many of the key individuals within the region. It covers Hecke algebras, corridor algebras, the Macdonald polynomial and its deviations, and their kin with different fields.

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**Example text**

A result of the second author [Pi] identified Schubert classes in the ho mogeneous space SO(2n,C)/U(n) with suitable Schur P-polynomials. In loc. cit. this identification used a geometric argument, namely an isomor phism 50(271, C)/C/(n) ~ S O ( 2 n - 1, C)/U(n- 1), and an identification of the Schubert classes for the latter Grassmannian with Schur P-functions. ) In the present paper we revisit the identification for the Schubert classes for SO(2n, C)/U(n) with Schur P-functions via a direct group-theoretic argument based on the calculus of divided differences of type D.

12. Assume that l(fi) < Z(A) + 1. (E) ^ 0, then D^- \Dx- is a horizontal strip. Proof. Suppose that Aj < /ij + i for some i. We can assume that for some 3

We end the paper by illustrating how the Schubert calculus developed here can be used to solve problems about enumeration of complex struc tures which satisfy some natural conditions of "partial overlaping" with a certain number of complex structures in general position in R 2 n . One of the applications leads to an interesting algebraic conjecture about homomorphisms between the cohomology ring of CS+ and that of the Grassmannian Gfe(C). Acknowledgements. S. N. (the "French connection" of this grant being A.