46 votes

Understanding a quip from Gian-Carlo Rota

I think Abdelmalek Abdesselam and William Stagner are completely correct in their interpretation of the words "Behind" and "one immutable source" as describing one theory, the theory of symmetric ...
24 votes

Understanding a quip from Gian-Carlo Rota

I think you are misinterpreting the quote. In the last sentence, the word "source" does not mean "source of these theories (K-theory, categories, group representations", but "source of the theory of ...
20 votes
Accepted

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
  • 108k
18 votes

Understanding a quip from Gian-Carlo Rota

Rota is not around anymore, so we can't go and ask him what he meant. My guess is that he is referring to the $\lambda$-ring structure of symmetric functions which is related to plethysm and the ...
16 votes
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Characterizing positivity of formal group laws

Given $\phi(x)\in\mathbb{R}[[x]]$, with $\phi(0)=1$, we have defined $g(x):=\int^x_0{dt\over \phi(t)}$, $f:=g^{-1}$ and $$F(x,y)=f\big(g(x)+g(y)\big)=\sum_{n=0}^\infty \psi_n(x) {y^n\over n!}\in\...
Pietro Majer's user avatar
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15 votes

universality of Macdonald polynomials

There are plenty of polynomials, and only a few are specializations of Macdonald polynomials. As a polynomial botanist, the following is a very incomplete family tree. A more comprehensive list of ...
Per Alexandersson's user avatar
15 votes
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An orbit of symmetric polynomials

Suppose $f(x,y,z)$ is symmetric (in the following, symmetric tout court always means "symmetric w.r.to the three variables $(x,y,z)$") . Then $\mathcal{L}f(x,y,z):=(x+y)(x+z)f(x-1,y,z)-x^2f(x,y,z)$ ...
Pietro Majer's user avatar
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15 votes
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Schur-Weyl duality and q-symmetric functions

As Sam Hopkins says, the category of all representations of $GL_n(\mathbb F_q)$ is too large to give what you want. Instead, let's consider the category of unipotent representations, i.e. those ...
Phil Tosteson's user avatar
14 votes
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Plugging $1-x$ into Schur polynomials

Let $t$ be an indeterminate. Let $\vartheta:\Lambda \rightarrow \Lambda[t]$ be the specialization (homomorphism) defined by $$...
Richard Stanley's user avatar
14 votes
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Todd polynomials

We have $$\log \sum_{k \ge 0} T_k t^k = \sum_{i=1}^n \log \frac{x_i t}{1 - e^{-x_i t}}$$ so if we write $$\log \frac{x_i t}{1 - e^{-x_i t}} = \log \sum_{k \ge 0} B_k^{+} x_i^k \frac{t^k}{k!} = \sum_{k ...
Qiaochu Yuan's user avatar
14 votes
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Is this generalized version of plethysm Schur positive?

No, this isn't true. Take $f = x^3+xy^2$, so that $f_{(12)} = y^3+x^2 y$. Then $$f+f_{(12)} = x^3+x^2 y + x y^2 + y^3=s_3$$ which is Schur positive. But $$h_2(x^3+xy^2, y^3+x^2 y) = x^6 + x^5 y + 3 x^...
David E Speyer's user avatar
13 votes
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Integer-valued power sums

The function $$ f : z \in \mathbb{C} \longmapsto \sum_{i} \frac{a_i}{1-a_iz} $$ is meromorphic on $\mathbb{C}$ and has integral Taylor coefficients. It follows from a theorem of Borel that such a ...
js21's user avatar
  • 7,199
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
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12 votes

Understanding a quip from Gian-Carlo Rota

Le teorie vanno e vengono ma le formule restano.--G.C. Rota. (The theories may come and go but the formulas remain.) Perhaps the Wiki on the Adams operation and "Formal groups, Witt vectors, and ...
11 votes
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A Schur positivity conjecture related to row and column permutations

I heard about this conjecture from Sara Billey at FPSAC, and I think I've got an argument. Let $F : \mathbb{C}[S_n] \to \mathbb{Z}[x_1, \ldots, x_N]^{S_N}$ be the linear map sending $w \mapsto p_{\rho(...
Brendan Pawlowski's user avatar
11 votes
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special values of symmetric functions at powers of $\frac1j$

As Gro-Tsen suggests in the comments, we have to expand the infinite product $$f(t)=\prod_j \left(1-\frac{t^8}{j^8}\right)=\prod_{j;\,w^4=1} \left(1-\frac{\omega t^2}{j^2}\right)=\prod_{w^4=1}\frac{\...
Fedor Petrov's user avatar
11 votes
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Using irreducible characters of the orthogonal group as basis for homogeneous symmetric polynomials

The coefficients do depend on $N$. A way to get around this and deal with "universal characters" was found by Koike and Terada (Young-diagrammatic methods for the representation theory of the ...
Richard Stanley'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
  • 108k
10 votes
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Schubert calculus expressed in terms of the cotangent space of the Grassmannians

The tangent space to the Grassmanian corresponds to the following representation of $U(r)\times U(n-r)$, call it $\rho$: it is the $r\times (n-r)$ matrices, with $U(r)$ acting on the left and $U(n-r)$ ...
Anton Mellit's user avatar
  • 3,572
10 votes

Bounded Degree in Ring of Symmetric Functions

Because we don't want to include power series like $$\frac{1}{1-\sigma_1}=1+\sigma_1+\sigma_1^2+\sigma_1^3+\cdots.$$ We barely even want power series at all; we're just using them as a notational ...
Tom Church's user avatar
  • 8,136
10 votes
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The vanishing of sum of coefficients: symmetric polynomials

Choose $n$ numbers $x_1,\dots,x_n$ for which all elementary symmetric polynomials are equal to 1 and substitute them to our $f_n$. We should get zero value for odd $n$. Well, what are these numbers? ...
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

Characterizing positivity of formal group laws

This is really just a comment. Your question is equivalent to the following: if we have a formal group law $$ F(x,y) = x + y + \sum_{i,j>0} a_{ij}x^iy^j \in \mathbb{Q}[[x,y]] $$ with $a_{1j}\geq 0$...
Neil Strickland'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
8 votes
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Details about plethysm

Of the three plethysms in 3), $h_n[h_2]$ is the simplest. A "modern" proof can be found for instance in Example A2.9 (page 449) of Enumerative Combinatorics, vol. 2. This was known to D. E. Littlewood ...
Richard Stanley's user avatar
8 votes

Bounded Degree in Ring of Symmetric Functions

One reason for considering $\Lambda$ is that many systems of symmetric polynomials form an honest $R$-basis of $\Lambda$, for example elementary symmetric, monomial symmetric, power symmetric, and ...
Friedrich Knop's user avatar
8 votes
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A Muirhead Like Inequality

$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\...
Iosif Pinelis's user avatar
8 votes
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Reference for Kakutani result on power sum bases of symmetric functions

I think Zagier must have been thinking of the following papers of Kakeya, instead of Kakutani Kakeya, S.: On fundamental systems of symmetric functions. I, II. Jap. J. Math.2, 69–80 (1925) ; 4, 77–...
Gjergji Zaimi's user avatar
8 votes
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Is there a Giambelli identity with dual representations?

Yes, your prediction is correct. The determinant identity in this case is theorem 3.5 in Division and the Giambelli Identity, by Wu and Yang (also published at Linear Algebra Appl. 406 (2005), 301-309)...
Gjergji Zaimi's user avatar

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