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20 votes

Why do we care about Schur Positivity

In algebraic combinatorics, conjectures that certain numbers are integers or are positive or are unimodal are really implicit challenges to find the hidden structure. And finding hidden structure is ...
Timothy Chow's user avatar
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15 votes
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Generating function for Schur polynomials

This is done in my paper The character generator of SU(n). I believe there was an essentially the same previous MO question, but I am unable to find it.
Richard Stanley's user avatar
12 votes
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Do you know an elegant proof for this expression for a Schur function?

I would like to suggest an interpretation using super symmetric functions. These are symmetric functions that are symmetric in two sets of variables $\{x_i\}$ and $\{y_j\}$ separately. They satisfy ...
Christopher Ryba's user avatar
11 votes

An identity related to partitions into $n$ parts and Schur polynomials

We can use the fact that $N(\lambda)=\left|\text{SSYT}(\lambda)\right|$, the number of semistandard Young tableaux of shape $\lambda$, and that $d!\cdot\left(\frac{\prod_{1\le i < j\le n}(\lambda_i ...
Gjergji Zaimi's user avatar
10 votes
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Is there a geometric interpretation of skew Schur functions?

This is discussed in Stanley's paper Some combinatorial aspects of the Schubert calculus. Corollary 3.7 says that under the natural isomorphism given by the Borel presentation of $H^*(G/P)$ which ...
Michael Joyce'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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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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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
7 votes
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Decompostion of hook schur function in terms of cauchy product of holonomic functions

It follows from the Cauchy identity and Exercise 7.43 of Enumerative Combinatorics, vol. 2, that $$ 1+ (u+t)\sum_{1\leq l\leq d} s_{l,1^{d-l}}(x)u^{l-1} t^{d-l} = \prod_i\frac{1+tx_i}{1-ux_i} $$...
Richard Stanley's user avatar
7 votes
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Cauchy identity, with sum restricted over partitions with first part $\leq n$

The sum (in any number of variables) is equal to the determinant $$\det(B_{j-i})_{1\le i,j\le n},$$ where $$B_i=\sum_{l=0}^\infty e_{l+i}(x)e_l(y),$$ and $e_l$ is the elementary symmetric function, ...
Ira Gessel's user avatar
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7 votes
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Is the appearance of Schur functions a coincidence?

As mentioned by Joel in the comments, I just posted this week on arxiv a paper that explains the relationship between some of these points using the geometric Satake correspondence (arxiv.org/abs/2310....
Antoine Labelle'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
6 votes

How to re-expand the sum of Schur function?

This is a special case of the generating function for Schur polynomials $$\sum_{\lambda}s_\lambda(x_1,\dots,x_m)s_\lambda(y_1,\dots,y_n)=\prod_{i=1}^m\prod_{j=1}^n\frac 1{1-x_iy_j}.$$ Take $x_1=\dots=...
Hjalmar Rosengren's user avatar
6 votes

Do you know an elegant proof for this expression for a Schur function?

Here is another argument. Let $\chi^{\mu/k}$ denote the skew character of the symmetric group $\mathfrak{S}_n$ corresponding to the skew shape $\mu/k$. Then by Pieri's rule, $$ \sum_{\mu-\lambda\ \...
Richard Stanley's user avatar
6 votes

An identity related to partitions into $n$ parts and Schur polynomials

I found a proof which I don't really like, but I'll share it. For two (real) diagonal matrices $A,B$, the Harish-Chandra-Itzykson-Zuber (HCIZ) integral is $$ I(A,B) = \int_{U(n)} e^{\rm{tr}(U^* A U B)...
sometempname's user avatar
6 votes

Most computationally efficient Littlewood-Richardson rule

I can refute Manivel's assertion that the transition equation is the most efficient way to multiply Schur (or Schubert) polynomials. I've implemented this, but later found a faster implementation ...
Matt Samuel's user avatar
  • 2,098
5 votes
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Harmonic flow on the Young lattice

Let us identify $s_\lambda$ with the character of the irreducible representation $S^\lambda$ of the symmetric group $S_n$ indexed by the partition $\lambda$. Then $$ s_\Box^{n-k}({\bf x}) \sum_{|\mu| =...
Christopher Ryba's user avatar
5 votes
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Is this simple symmetry of Littlewood-Richardson coefficients known?

I think you can understand this if you think of them as characters of $GL_m$-representations, i.e., $s_\lambda$ is the character of the irreducible representation which I'll denote by $V_\lambda$. The ...
Steven Sam's user avatar
  • 10.3k
5 votes

Significance of partition containment in representation theory of $\operatorname{GL}_n$

I am not quite sure what you are after. Maybe this: Let $V_\lambda$ be the polynomial irrep of $GL(n)$ corresponding to the partition $\lambda$. Then $\mu$ is contained in $\lambda$ if and only if $V_\...
Friedrich Knop'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
  • 1,499
4 votes

Finding Littlewood-Richardson coefficients without using identities

There are several approaches. Use linear algebra. Compute Schur polynomials (using the Jacobi-Trudi identity, say), and then use the fact that the coefficient of $s_\nu$ in the product $s_\lambda s_\...
Per Alexandersson's user avatar
4 votes
Accepted

Product of Schur functions

If $X = (u_1, u_2, \ldots, u_a, w_1, w_2, \ldots, w_c)$ and $Y = (v_1, v_2, \ldots, v_b, w_1, \ldots, w_c)$ with $u$'s, $v$'s and $w$'s disjoint, then $$s_{\lambda}(u,w) s_{\mu}(v,w) = \left( \sum_{\...
David E Speyer's user avatar
4 votes
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Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule

Please excuse that I answer with a link, I only have a phone right now. http://www.findstat.org/MapsDatabase/Mp00192/
Martin Rubey's user avatar
  • 5,573
4 votes

About the sum of rectangular power sums

Just adding that the expression for the Frobenius characteristic of $\theta_d$ is classically attributed to H. O. Foulkes, see e.g. Richard Stanley's EC2, Problem 7.88.
Sheila's user avatar
  • 41
4 votes
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Determinant connection between Schur polynomials and power sum polynomials

I don't know a fully general result, but your pattern for partitions $\lambda$ of length $\leq n$ with $n$-th entry $\lambda_{n}\geq n-1$ and with $n$ indeterminates persists: Theorem 1. Let $n$ be a ...
darij grinberg's user avatar
4 votes

Identities involving Littlewood–Richardson coefficients?

We looked into this with Greta and Damir in On the largest Kronecker and Littlewood–Richardson coefficients. Unfortunately, other than the Harris–Willenbring identity we didn't find much. For our ...
Igor Pak's user avatar
  • 16.8k
3 votes

Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule

Olga Azenhas' paper does not prove that for two fixed Young diagrams $\mu$ and $\lambda$, the Young diagrams $\nu$ with Littlewood-Richardson coefficients satisfying $c_{\mu,\nu}^\lambda>0$ form a ...
John Murray's user avatar
  • 1,070
3 votes
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About the sum of rectangular power sums

Talking to Sheila Sundram, as suggested in the comments, was a good idea. After some conversation, the following proof became apparent. I don't know of any proof already in the literature. Let $g$ ...
John Shareshian's user avatar
3 votes
Accepted

Sum of the ratios of Schur functions

I am skeptical of any "nice" structure while $x$ and $y$ remain so free. It might be somewhat reasonable to check things out under certain specializations. You may find some interesting quotients, ...
T. Amdeberhan's user avatar
3 votes

Integral of product of Schur functions

As a first step towards a solution, using the identity $$\int_{\mathcal{U}(N)} U_{\alpha a}U_{\alpha' a'}\bar{U}_{\beta b}\bar{U}_{\beta' b'}\,dU=\frac{1}{N^{2}-1}\bigl( \delta_{\alpha\beta}\delta_{ab}...
Carlo Beenakker's user avatar

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