对称函数空间的复合基
Compound basis for the space of symmetric functions
摘要：
本文的目的是为对称函数空间引入一个复合基。我们的基由Schur函数和$Q$ -函数的乘积组成。基本元素由分区索引。众所周知，Schur函数形成我们空间的规范正交基。一个自然的问题出现了。这两个基地是如何连接的？本文给出了这些基的转移阵的一些数值结果。特别地，我们将看到转移阵的行列式是2的幂。这并不令人惊讶。然而，显式公式涉及一个有趣的组合特征。我们的复合基来自于仿射李代数的基本表示的扭曲齐次实现。这篇笔记不是以数学文章的标准风格写的。它更像是一个演讲稿。特别是证明不给这里。细节和证据将在其他地方发表。
英文摘要：
The aim of this note is to introduce a compound basis for the space of symmetric functions. Our basis consists of products of Schur functions and $Q$-functions. The basis elements are indexed by the partitions. It is well known that the Schur functions form an orthonormal basis for our space. A natural question arises. How are these two bases connected? In this note we present some numerical results of the transition matrix for these bases. In particular we will see that the determinant of the transition matrix is a power of 2. This is not a surprising fact. However the explicit formula involves an interesting combinatorial feature. Our compound basis comes from the twisted homogeneous realization of the basic representation of the affine Lie algebras. This note is not written in a standard style of mathematical articles. It is more like a draft of a talk. In particular proofs are not given here. Details and proofs will be published elsewhere.
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