报告题目：The Ring of Isobaric Polynomials

报告摘要：

An isobaric polynomial  is a symmetric polynomial written on the ESP (Elementary Symmetric Polynomial ) basis.  The isobaric polynomials are indexed by partitions of the integers,  making them especially useful in Algebraic Combinatorics.  We choose two especially interesting linearly recursive sequences in this ring, the Generalized Fibonacci Polynomials (GFP) and the Generalized Lucas Polynomials, which when evaluated at for suitable values of the variables,  yield, respectively,  the classical Fibonacci sequence and the Lucas sequences.  GFP and GLP are also bases for the isobaric ring. We equip this ring with a convolution product,  which is a version of the Dirichlet Product in Arithmetic Function Theory,  that is the ring of maps from the Integers to the Complex Numbers. This is a ring which includes most of the interesting counting functions in mathematics; for example,the Euler Totient Function,the Möbius function, the Number of Divisors of n function, the Number of Prime Divisors of n function, etc. (there are uncountably many of them).  Multiplication in this ring is the Dirichlet Product.  There are two especially interesting subgroups in this ring; one is the Group of Multiplicative Arithmetic functions, and the other is the Group of Additive Arithmetic Functions.  These two subgroups have faithful representation in the Isobaric Ring;  namely, using,  respectively, the GFP’s and the GLP’s under the Convolution Product.  These representations have been useful in embedding  these two subgroups of the Arithmetic Function ring in their divisible closures.  It is a tantalizing problem to see if other parts of the Arithmetic Function ring can also be represented in the Isobaric ring.

报告时间：2016年12月1日下午3:00

报告地点：教学二楼伟德国际1946源自英国二楼报告厅

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