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 ...
Community wiki
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 ...
Community wiki
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
$$ s_k(a) := \frac{1}{\binom{n}{k}} S_k(a)$...
18
votes
Understanding a quip from Gian-Carlo Rota
Rota is not around anymore, so we can't go and ask him what he meant. My guess is that he is referring to the $\lambda$-ring structure of symmetric functions which is related to plethysm and the ...
Community wiki
16
votes
Accepted
Characterizing positivity of formal group laws
Given $\phi(x)\in\mathbb{R}[[x]]$, with $\phi(0)=1$, we have defined $g(x):=\int^x_0{dt\over \phi(t)}$, $f:=g^{-1}$ and $$F(x,y)=f\big(g(x)+g(y)\big)=\sum_{n=0}^\infty \psi_n(x) {y^n\over n!}\in\...
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 ...
15
votes
Accepted
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 $\mathcal{L}f(x,y,z):=(x+y)(x+z)f(x-1,y,z)-x^2f(x,y,z)$ ...
15
votes
Accepted
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 ...
14
votes
Accepted
Plugging $1-x$ into Schur polynomials
Let $t$ be an indeterminate. Let
$\vartheta:\Lambda
\rightarrow \Lambda[t]$
be the specialization (homomorphism) defined by
$$...
14
votes
Accepted
Todd polynomials
We have
$$\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 ...
14
votes
Accepted
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
$$h_2(x^3+xy^2, y^3+x^2 y) = x^6 + x^5 y + 3 x^...
13
votes
Accepted
Integer-valued power sums
The function
$$
f : z \in \mathbb{C} \longmapsto \sum_{i} \frac{a_i}{1-a_iz}
$$
is meromorphic on $\mathbb{C}$ and has integral Taylor coefficients. It follows from a theorem of Borel that such a ...
13
votes
Accepted
Cauchy identity in three sets of variables?
Yes, up to the hard problem of determining Kronecker coefficients. Let $\Delta^\lambda$ be the Schur functor for the partition $\lambda$ of $r$ and let $S^\lambda$ be the corresponding irreducible ...
12
votes
Understanding a quip from Gian-Carlo Rota
Le teorie vanno e vengono ma le formule restano.--G.C. Rota. (The theories may come and go but the formulas remain.)
Perhaps the Wiki on the Adams operation and "Formal groups, Witt vectors, and ...
Community wiki
11
votes
Accepted
A Schur positivity conjecture related to row and column permutations
I heard about this conjecture from Sara Billey at FPSAC, and I think I've got an argument. Let $F : \mathbb{C}[S_n] \to \mathbb{Z}[x_1, \ldots, x_N]^{S_N}$ be the linear map sending $w \mapsto p_{\rho(...
11
votes
Accepted
special values of symmetric functions at powers of $\frac1j$
As Gro-Tsen suggests in the comments, we have to expand the infinite product $$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{\...
11
votes
Accepted
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 ...
11
votes
Accepted
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(\...
10
votes
Accepted
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 $U(n-r)$ ...
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 ...
10
votes
Accepted
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? ...
9
votes
Accepted
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 $...
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 $a_{1j}\geq 0$...
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 ...
9
votes
Accepted
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 ...
8
votes
Accepted
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 ...
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 ...
8
votes
Accepted
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}{\...
8
votes
Accepted
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–...
8
votes
Accepted
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)...
Only top scored, non community-wiki answers of a minimum length are eligible
Related Tags
symmetric-functions × 325co.combinatorics × 143
rt.representation-theory × 89
reference-request × 41
algebraic-combinatorics × 39
symmetric-groups × 37
symmetric-polynomials × 37
schur-functions × 37
polynomials × 30
ac.commutative-algebra × 22
inequalities × 14
ag.algebraic-geometry × 13
young-tableaux × 13
linear-algebra × 10
plethysm × 9
nt.number-theory × 8
real-analysis × 8
determinants × 8
invariant-theory × 8
partitions × 8
orthogonal-polynomials × 8
special-functions × 7
enumerative-combinatorics × 7
ra.rings-and-algebras × 6
generating-functions × 6