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Published in:
Tensor Products Read chapter Read first chapter

2017 | OriginalPaper | Chapter

3. Symmetric Functions

Author : Alexey L. Gorodentsev

Published in: Algebra II

Publisher: Springer International Publishing

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Abstract

The symmetric group S n acts on the polynomial ring \(\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]\) by permutations of variables:
$$\displaystyle{ g\,f(x_{1},x_{2},\mathop{\ldots },x_{n}) = f\left (x_{g^{-1}(1)},x_{g^{-1}(2)},\ldots,x_{g^{-1}(n)}\right )\quad \forall \,g \in S_{n}. }$$
(3.1)
A polynomial \(f \in \mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]\) is called symmetric if g f = f for all g ∈ S n, and alternating if g f = sgn(g) ⋅ f for all g ∈ S n. The symmetric polynomials clearly form a subring of \(\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]\), whereas the alternating polynomials form a module over this subring, since the product of symmetric and alternating polynomials is alternating.

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previous chapter Tensor Algebras
next chapter Calculus of Arrays, Tableaux, and Diagrams
Footnotes
1
That is, consisting of at most n rows; see Example 1.​3 in Algebra I for the terminology related to Young diagrams.
 
2
Recall that the lexicographic order on \(\mathbb{Z}^{k}\) assigns (m 1, m 2, …, m k) > (n 1, n 2, …, n k) if the leftmost m i such that m i ≠ n i is greater than n i.
 
3
The coefficient of every monomial changes sign under the transposition of any two variables.
 
4
Recall that we use the notation ( f(i, j)), where f(i, j) is some function of i, j, for the matrix having f(i, j) at the intersection of the ith row and jth column.
 
5
That is, obtained from one another by reflection in the main diagonal.
 
6
Over an arbitrary (even noncommutative) ring with unit.
 
7
That is, f(e 1,e 2,…,e n) ≠ 0 in \(\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]\) for every \(f \in \mathbb{Z}[t_{1},t_{2},\ldots,t_{n}]\).
 
8
See Sect. 12.2.3 of Algebra I.
 
9
That is, the cardinality of the stabilizer of the permutation of cyclic type λ under the adjoint action of the symmetric group; see Example 12.16 of Algebra I.
 
10
See Corollary 4.​3 on p. 94.
 
11
Recall that we write ℓ(λ) for the number of rows in a Young diagram λ and call it the length of λ.
 
12
The first form a ring, and the second form a module over this ring.
 
13
For n = 0, we put \(\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]\) equal to \(\mathbb{Z}\).
 
14
Recall that the n th cyclotomic polynomial \(\Phi _{n}(x) =\prod (x-\zeta )\) is the monic polynomial of degree φ(n) whose roots are the primitive nth roots of unity \(\zeta \in \mathbb{C}\). (See Sect. 3.5.4 of Algebra I.)
 
15
See Proposition 9.​4 from Algebra I.
 
Literature
[DK]
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.MathSciNetCrossRef
[Fu]
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.MATH
[FH]
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.MATH
[Mo]
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
Metadata
Title
Symmetric Functions
Author
Alexey L. Gorodentsev
Copyright Year
2017
Publisher
Springer International Publishing
Book
Algebra II

Print ISBN: 978-3-319-50852-8

Electronic ISBN: 978-3-319-50853-5

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-50853-5_3

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