The Schubert polynomial [G.sub.w] for a permutation w in the

symmetric group [S.sub.n] is defined as follows.

Equation (8) allows defining a maximum

symmetric group (G) of the image:

Our interest in this result stems from the crucial role it plays in Quillen's method [8] for proving homological stability of the

symmetric groups. Following Hatcher and Wahl [5], we shall formulate and prove a slightly more general theorem so that it can be applied in the proof of homological stability of wreath-product groups.

1 INTRODUCTION: The

symmetric group Sn is defined over the regular figure n-gon with order n!

As the remedy module following OPM, SHM takes into consideration the wavelengths being allocated to the

symmetric group of the HC construction request of the flow, and it maintains the wavelength sequences of certain intermediate nodes in real time.

The

symmetric group [G.sub.n] on {1, ..., n} is generated by {[s.sub.1], ..., [s.sub.n-1]}, where [s.sub.i] is the permutation interchanging i and i + 1, and fixing all other elements.

The

symmetric group [??] on {1,..., n} is generated by {[s.sub.1],..., [s.sub.n-1]}, where [s.sub.i] is the permutation interchanging i and i + 1, and fixing all other elements.

So the

symmetric group of scaling transformations is determined:

If |X| = n, [S.sub.X] is denoted by [S.sub.n] and called the

symmetric group of degree n.

There is a Schubert polynomial [S.sub.w] for each permutation w in the

symmetric group [S.sub.[infinity]] = [union] [S.sub.m], satisfying the following two rules.

The 16 papers from the latest Osaka conference on Schubert calculus consider such topics as consequences of the Lakshmibai-Sandhya theorems: the ubiquity of permutation patterns in Schubert calculus and related geometry, stable quasi-maps to holomorphic symplectic quotients, tableaux and Eulerian properties of the

symmetric group, generalized (co)homology of the loop spaces of classical groups and the universal factorial Schur P- and Q-functions, and character sheaves on exotic symmetric spaces and Kostka polynomials.

The

symmetric group [[??].sub.n] acts on the ring of polynomials in [x.sub.n] = {[x.sub.1],...,[x.sub.n]}, with coefficients in other alphabets.