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Differential Operator Specializations of Noncommutative Symmetric Functions | | Wenhua Zhao
; | | Date: |
7 Sep 2005 | | Subject: | Combinatorics; Quantum Algebra MSC-class: Primary: 05E05, 14R10, 16S32; Secondary: 16W20, 16W30 | math.CO math.QA | | Abstract: | Let $K$ be any unital commutative $Q$-algebra and $z=(z_1, ..., z_n)$ commutative or noncommutative free variables. Let $t$ be a formal parameter which commutes with $z$ and elements of $K$. We denote uniformly by $kzz$ and $kttzz$ the formal power series algebras of $z$ over $K$ and $K[[t]]$, respectively. For any $alpha geq 1$, let $cDazz$ be the unital algebra generated by the differential operators of $kzz$ which increase the degree in $z$ by at least $alpha-1$ and $ ataz $ the group of automorphisms $F_t(z)=z-H_t(z)$ of $kttzz$ with $o(H_t(z))geq alpha$ and $H_{t=0}(z)=0$. First, for any fixed $alpha geq 1$ and $F_tin ataz$, we introduce five sequences of differential operators of $kzz$ and show that their generating functions form a cNcs (noncommutative symmetric) system (cite{GTS-I}) over the differential algebra $cDazz$. Consequently, by the universal property of the cNcs system formed by the generating functions of certain NCSF’s (noncommutative symmetric functions) first introduced in cite{G-T}, we obtain a family of Hopf algebra homomorphisms $cS_{F_t}: cNsf o cDazz$ $(F_tin ataz)$, which are also grading-preserving when $F_t$ satisfies certain conditions. Note that, the homomorphisms $cS_{F_t}$ above can also be viewed as specializations of NCSF’s by the differential operators of $kzz$. Secondly, we show that, in both commutative and noncommutative cases, this family $cS_{F_t}$ (with all $ngeq 1$ and $F_tin ataz$) of differential operator specializations can distinguish any two different NCSF’s. Some connections of the results above with the quasi-symmetric functions (cite{Ge}, cite{MR}, cite{St2}) are also discussed. | | Source: | arXiv, math.CO/0509134 | | Services: | Forum | Review | PDF | Favorites | |
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