Ring of symmetric functions: Difference between revisions

Revision as of 20:27, 13 December 2021

Revision as of 11:17, 9 November 2022

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:<math>P(t) = -t\frac d{dt}\log(E(-t)) = t\frac d{dt}\log(H(t)),</math>

which amounts to the same, but requires that ''R'' contain the rational numbers, so that the logarithm of power series with constant term&nbsp;1 is defined (by <math>\textstyle\log(1-tS) = -\sum_{i>0} \frac1i(tS)^i</math>).

== Specializations ==

Let <math>\Lambda</math> be the ring of symmetric functions and <math>R</math> a commutative algebra with unit element. An algebra homomorphismus <math>\varphi:\Lambda\to R,\quad f\mapsto f(\varphi)</math> is called a ''specialization''.<ref name="StanleyFomin">{{cite book|last1=Stanley|first1=Richard P.|last2=Fomin|first2=Sergey P.|title= Enumerative Combinatorics|volume=2|publisher=Cambridge University Press}}</ref>

Example:

* Given some real numbers <math>a_1,\dots,a_k</math> and <math>f(x_1,x_2,\dots,)\in \Lambda</math>, then the substitution <math>x_1=a_1,\dots,x_k=a_k</math> and <math>x_j=0,\forall j>k</math> is a specialization.

* Let <math>f\in \Lambda</math>, then <math>\operatorname{ps}(f):=f(1,q,q^2,q^3\dots,)</math> is called ''principal specialization''.

==See also==