Abstract
The symmetric group S n acts on the polynomial ring \(\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]\) by permutations of variables:
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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Notes
- 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.
References
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
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Gorodentsev, A.L. (2017). Symmetric Functions. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_3
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