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20 votes

Maclaurin's inequality on elementary symmetric polynomials of arbitrary real numbers

I've just completed a short paper establishing a positive answer to this question here (see also this blog post). In terms of the elementary symmetric means $$ s_k(a) := \frac{1}{\binom{n}{k}} S_k(a)$...
Terry Tao's user avatar
  • 112k
15 votes
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Construction of a symmetric polynomial in the roots that acts like the discriminant

In characteristic not equal to $2$, the discriminant is optimal. In characteristic $2$, the polynomial $\prod_{i<j} (x_i+x_j)$ works and has degree $\binom{n}{2}$. Proof: Let $f(x_1, x_2, \ldots, ...
David E Speyer's user avatar
14 votes

Symmetric polynomials that detect positivity

The elementary symmetric functions $1=e_0,e_1,\dots,e_n$ will suffice. Clearly $e_j>0$ if each $a_i>0$. Conversely, let $P(x)= \prod_{i=1}^n(x+a_i)$. If some $a_i<0$ and all $e_j>0$ then $...
Richard Stanley's user avatar
14 votes
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When the Littlewood-Richardson rule gives only irreducibles?

The answer is Yes, but this requires some elaboration. Knutson-Tao-Woodward prove Fulton's conjecture in $\S$6.1. In principle, you can follow the approach by De Loera-McAllister or Mulmuley-...
Igor Pak's user avatar
  • 16.8k
13 votes
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Cauchy identity in three sets of variables?

Yes, up to the hard problem of determining Kronecker coefficients. Let $\Delta^\lambda$ be the Schur functor for the partition $\lambda$ of $r$ and let $S^\lambda$ be the corresponding irreducible ...
Mark Wildon's user avatar
11 votes
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Decomposition of a tensor product of representations of $\mathrm{GL}_l(\mathbb{C})$ and decomposition of Littlewood-Richardson numbers?

This identity can be deduced from the hive model of Littlewood-Richardson coefficients, which Allen Knutson and I introduced in Knutson, Allen; Tao, Terence, The honeycomb model of (\text{GL}_n(\...
Terry Tao's user avatar
  • 112k
10 votes

Polynomial inequality $n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3$

Take $n=3k$, $2k$ variables equal to $3$ and $k$ variables equal to $-5$ for large $k$. Then $\sum a_i=k>0$, and $\sum_{i<j<k} a_ia_ja_k=\frac16 (\sum a_i)^3+O(k^2)=\frac{k^3}6+O(k^2)>0$ ...
Fedor Petrov's user avatar
10 votes
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Formula expressing symmetric polynomials of eigenvalues as sum of determinants

This is not a reference, but a short proof. We use the following (probably known, but see later) lemma on representing a symmetric tensor as a linear combination of rank-1 symmetric tensors. Lemma. ...
Fedor Petrov's user avatar
9 votes
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Inequality with symmetric polynomials

This looks like a better fit for Math Stackexchange, because it's the kind of thing one learns from Olympiad problem books . . . One standard approach that has not been mentioned yet: We may assume $...
Noam D. Elkies's user avatar
9 votes

Formula expressing symmetric polynomials of eigenvalues as sum of determinants

Concerning the reference request: Several text books [1,2] give the theorem and proof for elementary symmetric polynomials $s_k=$ sum of all $k\times k$ principal minors of the $n\times n$ matrix. ...
Carlo Beenakker's user avatar
9 votes

Generalization of symmetric functions

Let $w\in S_n$ (the symmetric group) have cycle type $\lambda =(\lambda_1,\dots, \lambda_\ell)\vdash n$, where $\ell=\ell(\lambda)$ is the length (number of nonzero parts) of $\lambda$. Then the ...
Richard Stanley's user avatar
9 votes
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Nonnegativity locus of Schur polynomials

The answer to the main question is affirmative. The crucial result is due to M. Aissen, I. J. Schoenberg, and A. Whitney, J. Analyse Math. 2 (1952), 93—103. For further details see the solution to ...
Richard Stanley's user avatar
9 votes
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Can the ring of symmetric polynomials be generated by powers of symmetric polynomials of degree 1?

Yes. For variables $x_1,\dots, x_k$, we have $$ \sum_{J \subseteq \{1,\dots, k \} } (-1)^{ k- |J|} \left(\sum_{j \in J} x_j\right)^k = k! x_1 \dots x_k $$ so that $$ e_k = \sum_{(i_1; \dots; i_k )\in ...
Will Sawin's user avatar
  • 141k
9 votes
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Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials

Since $x \, \partial_{x} \, x^k = k \, x^k$, the operator $L$ just records the total degree of a polynomial. For Schur (and Jack, Macdonald, $\dots$) polynomials it thus acts by $$L \, s_\lambda(x_1,\...
Jules Lamers's user avatar
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7 votes
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Maximize $L^p$ norm over sphere

Proof for the case $p$ odd: The Lagrange-multiplier equation yields $$ p\lambda_i^{p-1}+a\lambda_i+b=0 $$ for some $a,b \in \mathbb{R}$. If $p>1$ is odd the LHS is a strictly convex function in $\...
user35593's user avatar
  • 2,286
7 votes
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Schur positivity of a polynomial

Given $f_1,\dots,f_p$ and $d\geq \max f_i$, a necessary and sufficient condition is that all zeros of the polynomial $\sum x^{f_j}$ are real. See Enumerative Combinatorics, vol. 2, Exercise 7.91. Note....
Richard Stanley's user avatar
7 votes

Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials

The answer for $L_1=\sum_ix_i^2\partial_i$ can be derived in a rather straightforward way (I changed your convention a little bit to match the usual formulas for Virasoro algebra). Namely, use the ...
Vladimir Dotsenko's user avatar
7 votes
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The coefficients of the Jack polynomials are polynomials in the Jack parameter

Following up on the suggestion of LSpice, to remove this from the "unanswered queue": the combinatorial formula in Wikipedia, due to Knop and Sahi, is a polynomial in $\alpha$.
Carlo Beenakker's user avatar
6 votes

Examples of specializations of elementary symmetric polynomials

The specialization $\mathcal{S} = \{1, -1, \frac12, -\frac12, \frac13, -\frac13, \ldots \}$, in the limit $n\to \infty$, produces the Weierstrass factorization $$ \frac{\sin \pi z}{\pi z} = \prod_{n\...
Harry Richman's user avatar
6 votes

Polynomial inequality of sixth degree

I have found the following identity, which solves my problem for $k=\frac{13}{5}.$ $$4\prod_{cyc}(a+b)^2\left(\sum_{cyc}\frac{2a^2+bc}{(b+c)^2}-\frac{9}{4}\right)=\frac{1}{3}\left(\sum_{cyc}(2a^3-a^2b-...
Michael Rozenberg's user avatar
6 votes

Polynomial inequality of sixth degree

Your inequality fails to hold when e.g. $k=25999/10000=2.6-10^{-4}$ and $(a,b,c)=(97661/65536,-5/3,-1)$. Indeed, the smallest value for which your inequality holds is $13/5=2.6$. Here is a proof by ...
Iosif Pinelis's user avatar
5 votes
Accepted

Frobenius algebras from symmetric polynomials

In this commutative situation, a Frobenius algebra is the same as an artinian Gorenstein ring. In general, if $\theta_1,\dots,\theta_n$ are homogeneous elements of positive degree of $A=K[x_1,\dots,...
Richard Stanley's user avatar
5 votes
Accepted

Normalization of Jack polynomial integral-scalar product?

Yes, my friend. Take $J_\lambda^{(\alpha)}$ in the J-normalization. Let $n$ be the number of variables (which for you is $2$). Let $\lambda'$ denote the conjugate partition to $\lambda$. Define $$ C_\...
E.tsukerman's user avatar
5 votes
Accepted

Singular locus of zero set of elementary symmetric polynomial

Notice that the vanishing of $\frac{\partial}{\partial x_1} \sigma_{m,r}, \cdots, \frac{\partial}{\partial x_m} \sigma_{m,r}$ implies the vanishing of $\sigma_{m,r}$ since $$\sigma_{m,r}=\frac{1}{r}\...
Gjergji Zaimi's user avatar
5 votes
Accepted

Proof of a combinatorial identity for a sum over partitions of sets giving rise to a symmetric polynomial?

The notation and framing of the problem are not optimal, because the $n_i$'s being integers is a bit of a distraction. Let $[k]:=\{1,\ldots,k\}$, and let ${\rm Part}_k$ be the set of set partitions of ...
Abdelmalek Abdesselam's user avatar
5 votes

Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials

The answer by Jules Lamers considers $k=1$. In the comments to that answer, people have mentioned that the case $k=0$ has been solved. Namely, for a given partition $\lambda$ let Del($\lambda$) be the ...
thedude's user avatar
  • 1,519
4 votes

When is a linear combination of the elementary symmetric polynomials reducible?

Suppose that $f=p_1 p_2\cdots p_k$, where each $p_i$ is irreducible. Since $f$ is a symmetric function, for every $w\in S_n$ and every $i$ we must have $w\cdot p_i = cp_j$ for some $j$, where $c$ is a ...
Richard Stanley's user avatar
4 votes

Inequality with symmetric polynomials

This is easily proven using the Rearrangement Inequality, which says that if we have two sequences of reals, and we are to pair them up in such a way as to maximize the sum of the products of the ...
R.P.'s user avatar
  • 4,745
4 votes

Inequality with symmetric polynomials

Follows from Hölder's inequality (p=6, q = 6/5): $ab^5 + ba^5 \le (a^6+b^6)^{1/6} (b^6+a^6)^{5/6}$
jjcale's user avatar
  • 2,768
4 votes

Examples of specializations of elementary symmetric polynomials

Under the specialization $\mathcal{S} = \{1, \frac12, \frac13, \ldots, \frac1n \}$, the corresponding products to not converge to a limit as $n\to\infty$ since the harmonic series diverges. However, ...
Harry Richman's user avatar

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