Elementary symmetric polynomial: Difference between revisions

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'''Lemma'''. The leading term of {{math|''e''_''λ''^t (''X''₁,…,''X''_''n'')}} is {{math|''X''^''λ''}}.

:''Proof''. To get theThe leading term of the product is the product of the leading terms of each factor (this is true whenever one mustuses selecta [[monomial order]], like the lexicographic order used here), and the leading term inof eachthe factor {{math|''e''_''i''(''X''₁,…,''X''_''n'')}}{{why|date=March 2013}}, which is clearly {{math|''X''₁''X''₂…''X''_''i''}}, and multiply these together. To count the occurrences of the individual variables in the resulting monomial, fill the column of the Young diagram corresponding to the factor concerned with the numbers 1…,{{mvar|''i''}} of the variables, then all boxes in the first row contain 1, those in the second row 2, and so forth, which means the leading term is {{math|''X''^''λ''}} (its coefficient is 1 because there is only one choice that leads to this monomial).

Now one proves by induction on the leading monomial in lexicographic order, that any nonzero homogeneous symmetric polynomial {{mvar|''P''}} of degree {{mvar|''d''}} can be written as polynomial in the elementary symmetric polynomials. Since {{mvar|''P''}} is symmetric, its leading monomial has weakly decreasing exponents, so it is some {{math|''X''^''λ''}} with {{mvar|''λ''}} a partition of {{math|''d''}}. Let the coefficient of this term be {{mvar|''c''}}, then {{math|''P'' − ''ce''_''λ''^t (''X''₁,…,''X''_''n'')}} is either zero or a symmetric polynomial with a strictly smaller leading monomial. Writing this difference inductively as a polynomial in the elementary symmetric polynomials, and adding back {{math|''ce''_''λ''^t (''X''₁,…,''X''_''n'')}} to it, one obtains the sought for polynomial expression for {{math|''P''}}.