Questions tagged [schur-functions]
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97 questions
10
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3
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A question on the plethysm of complete symmetric functions
Based on some small calculations in SageMath, I conjectured that the Schur expansion of $h_n[h_k]$ contains $s_{(k,k,...,k)}$ if and only if $k$ is even. For example, this is easily seen when $n=2$. ...
- 419
3
votes
0
answers
112
views
Link between Schur polynomials and generators of the Virasoro algebra
I am currently studying Schur polynomials in the context of a representation of the Virasoro algebra for bosons (with central charge $c=1$). The generators of the algebra are denoted $L_k^{(n)}$, in ...
- 131
12
votes
2
answers
456
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Is $(1+z_1 + \dotsb +z_1^m)^p \dotsb (1+z_n + \dotsb +z_n^m)^p$ Schur positive for large enough $p$?
I would like to know if the symmetric polynomial $(1+z_1 + \dotsb +z_1^m)^p \dotsb (1+z_n + \dotsb +z_n^m)^p$ is Schur positive for large enough $p$. Here, how large the $p$ is can depend on $n$ and $...
- 121
2
votes
0
answers
73
views
linear recurrence for Kostka-Foulkes polynomials
Let $\lambda$ and $\mu$ be weights of some simply-connected reductive group $G$ over $\mathbb{C}$, viewed as elements of the affine Weyl group of $G$. Then one of the many ways to define the ...
- 513
7
votes
0
answers
163
views
Is there a "Weyl Character Formula" for Jacobi Polynomials
Let $R$ be a root system, $R^+$ a choice of positive roots, $W$ the Weyl group, $\Lambda$ its weight lattice, $\Lambda^+$ the cone of dominant weights, and $\rho = \frac{1}{2} \sum_{\alpha \in R^+} \...
- 285
4
votes
0
answers
156
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Reference for fact about flags of vexillary permutations
Vexillary permutations are an important family of permutations in Schubert calculus. There are several definitions, for example that they avoid the pattern 2143.
Recall the Lehmer code of a ...
- 2,009
7
votes
1
answer
424
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Jacobi-Trudi-like identity with dual characters
If $\lambda$ is a partition with at most $n$ parts, let $s_\lambda$ be the corresponding Schur polynomial in $n$ variables $x_1,\ldots,x_n$. In particular, for $a \geq 0$, $s_{a}$ is the complete ...
- 1,282
7
votes
2
answers
608
views
Proving an identity for flagged Schur without use of determinants?
In proposition 3 of Determinantal transition kernels for some interacting particles on the line, Dieker and Warren prove the following identity: consider vector $a:=(a_1,\dotsc,a_N)$ and kernels
$$\...
- 5,599
4
votes
1
answer
179
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Are the minors of this Hadamard product Schur positive?
Let $h_i (x)$ denote the complete symmetric function of degree $i$ in some set of variables $x = (x_1 , x_2 , \dots)$. Then the minors of the Toeplitz matrix $T (x) = \left(h_{i-j} (x) \right)_{i,j}$ ...
- 563
6
votes
3
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658
views
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 operators $L_k=\sum_i x_i^k\frac{\partial}{\partial x_i}$, with integer $k$, take symmetric polynomials into symmetric polynomials.
Is it known how to write the result of the application of $L_0$, ...
- 1,549
3
votes
0
answers
249
views
Factorization of symmetric polynomials
Let $\Lambda_n$ be the algebra of all symmetric polynomials in $n$ variables, which we also consider as an infinite-dimensional vector $\mathbb{Q}$-space, whose basis is the Schur polynomials.
The ...
- 656
4
votes
1
answer
304
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Significance of partition containment in representation theory of $\operatorname{GL}_n$
I'm working on a Schur positivity problem and I came across a series of Schur polynomials (all in $n$ variables) whose indexing partitions are (conjecturally) contained in one of some given set of ...
- 61
12
votes
1
answer
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Is the appearance of Schur functions a coincidence?
The Schur functions are symmetric functions which appear in several different contexts:
The characters of the irreducible representations for the symmetric group (under the characteristic isometry).
...
- 193
3
votes
0
answers
143
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"Sufficiently fat" Schur polynomial times Schubert polynomial is Schubert-positive
It's still open to find a combinatorial proof that for integers $k>0$ and $m>0$, a permutation $u\in S_\infty$ with its final descent at position $m$, and a partition $\lambda$ that
$$\mathfrak{...
- 2,448
5
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0
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166
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Representation-theoretic interpretation of double Schur polynomials
The Schur polynomials
$$s_\lambda(x_1, \ldots, x_n) = \frac{|x_i^{\lambda_j+n-j}|_{1\le i,j\le n}}{|x_i^{n-j}|_{1\le i,j\le n}}$$
naturally appear as polynomial representatives for Schubert classes in ...
- 4,097
4
votes
1
answer
577
views
Is this simple symmetry of Littlewood-Richardson coefficients known?
Let $\lambda$ be a partition with at most $p$ parts, let $\mu$ be a partition with at most $q$ parts, and let $\nu$ be a partition with at most $p+q$ parts. Let $m\geq \nu_1$ be an integer. We denote ...
- 2,448
9
votes
1
answer
390
views
The convex hull of Schur polynomial evaluations
Let $r\leq n$ and $d$ be positive integers. A probability vector is a vector of non-negative entries that sum to 1. For each probability vector $\lambda$ of length $n$, let
$$s(\lambda)=(\dim[\pi] \...
- 970
4
votes
2
answers
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About Cauchy identity for Schur polynomials
(This was originally posted here, https://math.stackexchange.com/questions/4687466/cauchy-identity-for-schur-functions, and I am reposting it here as it seems to be more appropriate.)
PRELIMINARY.
The ...
- 121
0
votes
1
answer
460
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Dose density matrix with off-diagonal elements equal to zero has maximum von-Neumann entropy?
von-Neumann entropy
I know von-Neumann entropy on density matrix $S=-{\rm Tr}(\rho \ln\rho)$ is similar to Shannon entropy $S=-\sum_i p_i\ln p_i$ in classical mechanics. And I want to get Bose-...
- 3
5
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0
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188
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Sum of Schur functions associated to self-conjugate partitions
The $\tau$-function $H^\circ \big(t ;\vec{x} \big)$ associated with counting simple Hurwitz numbers is the formal power series
\begin{equation}
(\dagger) \quad H^\circ \big(t ;\vec{x} \big) \, =
\,
\...
- 2,257
2
votes
1
answer
430
views
Identities involving Littlewood–Richardson coefficients?
I am not aware of that many identities that involve several Littlewood–Richardson coefficients.
One recent identity, is a generating function as sum of squares of LR-coefficients,
due to Harris and ...
- 16.1k
8
votes
1
answer
446
views
Harmonic flow on the Young lattice
Let me begin with some preliminary concepts: A positive real-valued function $\varphi: P \rightarrow \Bbb{R}_{>0}$ on a locally finite, ranked poset $(P, \trianglelefteq)$ is harmonic if
$\varphi(\...
- 2,257
4
votes
1
answer
421
views
Nonnegativity locus of Schur polynomials
Let $a_1,\ldots,a_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such ...
- 6,886
16
votes
0
answers
616
views
Identity involving Schur polynomials, binomial coefficients and contents of partition
Let $C_{\lambda,\mu}$ be the coefficients defined as
$$ s_\lambda\left(\frac{x_1}{1-x_1},...,\frac{x_N}{1-x_N}\right)=\sum_{\mu\supset \lambda}C_{\lambda\mu}s_\mu(x_1,...,x_N),$$
where $s$ are the ...
- 2,572
10
votes
1
answer
883
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Determinant connection between Schur polynomials and power sum polynomials
Let $f_i=f_i(x_1,x_2,\ldots, x_n),i=0,1,2, \ldots $ be a family of symmetric polynomials. For the partition $\lambda=(\lambda_1,\lambda_2, \ldots, \lambda_n)$ consider the determinant
$$
D_\lambda(f)...
- 656
11
votes
1
answer
904
views
Generating function for Schur polynomials
Consider the generating function
$$
G_n(x_1,x_2,\ldots,x_n, t_1,t_2,\ldots,t_n) =\sum_{\lambda}s_{\lambda}(x_1,x_2,\ldots, x_n) t_1^{\lambda_1}t_2^{\lambda_2} \cdots t_n^{\lambda_n},
$$
where the sum ...
- 656
2
votes
1
answer
192
views
How to re-expand the sum of Schur function?
Consider next sum
\begin{eqnarray}
\label{PF_spindef}
Z = \sum_{r=0}^{N N_f} h^{2r} \ Q(r) .
\end{eqnarray}
and
\begin{equation}
Q(r) \ = \ \sum_{\sigma \vdash r} s_{\sigma}(1^{N_f}) \
s_{\sigma}...
- 113
12
votes
2
answers
510
views
Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule
The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...
- 431
7
votes
2
answers
308
views
About the sum of rectangular power sums
Let $n \geq 1$ be an integer and consider the symmetric function
$$D_n = \sum_{d|n} p_d^{n/d},$$
where $p_{d}$ are the power-sum symmetric functions.
It can be checked up to $n=35$ that the symmetric ...
- 3,793
1
vote
0
answers
82
views
LGV scheme: Any situations where the weights shift differently for each path?
In Cylindric partitions, Proposition 1, Gessel and Krattenthaler prove a formula for lattice paths on a cylinder
In our particular problem, we again have paths $((P_{1},k_{1}),...,(P_{r},k_{r}))$ but ...
- 5,599
4
votes
0
answers
148
views
Tensors of minimal rank in Schur modules $S_{\lambda}V \subset V^{\otimes |\lambda|}$
It is well known that for a vector space $V$ with $\dim(V)=n+1$ the $GL(V)-$module $V^{\otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{\lambda}V$, where $\...
- 1,353
4
votes
1
answer
288
views
proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions"
I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...
- 41
2
votes
0
answers
117
views
Double Schur function expansion
In literature, I have seen the weighted Hurwitz number $N_{g,n}(d_1 , d_2 \ldots , d_n)$ which are symmetric number and they can be written as double Schur function expansion.
\begin{align} \label{eq:...
- 695
4
votes
2
answers
379
views
LGV scheme for lattice paths that move in non-unit spatial positive steps
In the Lindström–Gessel–Viennot lemma (LGV) applied to the $Z^2$-lattice paths are taken to move in unit spatial-steps in unit time (see here).
What do we mean by "time"? In the language of ...
- 5,599
15
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2
answers
940
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Do you know an elegant proof for this expression for a Schur function?
I know that the identity
$$
s_\mu = \sum_{\mu-\lambda \text{ is a horizontal strip}} \;\sum_{\alpha\vdash|\lambda|} \frac{\chi^\lambda_\alpha}{z_\alpha} \prod_i(p_i-1)^{a_i}
$$
holds.
Here $\alpha=1^{...
- 5,894
6
votes
0
answers
281
views
Derivations for symmetric functions
A symmetric function is a formal power series in infinitely many variables $x_1,x_2,\dots$ invariant under the permutation of variables (as opposed to a polynomial). Let $\Lambda$ denote the algebra ...
- 2,009
0
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0
answers
114
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Theorem 5.3 ([Okounkov-01]) in Borodin and Gorin's lecture note
In this lecture note: https://arxiv.org/pdf/1212.3351.pdf, Theorem 5.3(P28):
Suppose that the $\lambda \in \mathbb{Y}$ is distributed according to the Schur measure $\mathbb{S}_{\rho_1; \rho_2}$. ...
- 308
1
vote
0
answers
317
views
What is an orthogonal form?
I am reading the article: Algebro - Geometric applications of Schur S- and Q-polynomials
On page 179, they said about an orthogonal form $\psi$ on $W$. There is no definition of an orthogonal form on ...
- 320
1
vote
1
answer
318
views
Schur polynomials with zeros in an infinite geometric progression
Let $a_1,...,a_n\in \overline{\mathbb{Q}}^\times$ be algebraic numbers, such that $\frac{a_i}{a_j}$ is not a root of unity for $i\neq j$. Furthermore, let $m\in\mathbb{N}$, $m>1$.
Question: Is ...
- 171
3
votes
1
answer
456
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Finding Littlewood-Richardson coefficients without using identities
The Littlewood-Richardson coefficients $C^{R}_{QP}$ for some partitions $R, Q, P$ can usually be dealt with using identities like for example $$C^{R}_{QP} = 0 \quad \text{ if } \quad|R| \neq |Q| + |P|...
1
vote
0
answers
162
views
Calculation of complete homogeneous symmetric functions [closed]
Say you have a complete homogeneous symmetric function $$h_4 = \sum_{1\leq i \leq j \leq k \leq l}q^{-i}q^{-j}q^{-k}q^{-l},$$ where $i = 1, 2, 3, \ldots$. There are 7 cases to consider, given by
$$...
3
votes
0
answers
153
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Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)
For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...
- 131
2
votes
0
answers
148
views
Restricted Cauchy identity
Is there some reference for sums like:
$$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)t^{|\nu|}$$
$$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)\cdot|\nu|$$
(summation ...
- 645
4
votes
1
answer
299
views
Decomposing tensor powers of the fundamental representation of exceptional Lie algebras
For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, ...
- 845
2
votes
0
answers
87
views
Annihilator of the of the generating function not holonomic
The following is a generating function in $x,h$ with infinite parameters
$q_1,q_2\ldots,$ and $w_1, w_2,\ldots$.
$$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\...
- 695
1
vote
0
answers
44
views
Extension of definition of Holonomic closure
My question is about finding the annihilator of a series. Let me begin with what is known and then ask my question. Let $s_d(\frac{q_1}{h},\ldots )$ denote schur function for partition $\lambda =[d]$ ...
- 695
2
votes
0
answers
102
views
Schur function on unit circles
Define $T^d$ as following
$$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{C}^{d}\mid |t_i|= 1 \mbox{ for all } i\right\}
$$
For any partition $\lambda\vdash n$,The Schur function is defined
$$
\...
- 1,533
13
votes
1
answer
445
views
Is there a Giambelli identity with dual representations?
For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition.
...
- 164k
10
votes
1
answer
387
views
Integral of product of Schur functions
Schur functions are irreducible characters of the unitary group $\mathcal{U}(N)$. This implies the integration formulae
$$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^...
- 2,572
6
votes
0
answers
294
views
Macdonald's "Symmetric Functions and Hall Polynomials" Section 1.5 Example 9
I'm trying to follow Example 9 in Section 1.5 of the 2nd edition of Macdonald's book "Symmetric Functions and Hall Polynomials". I have trouble with understanding some points.
Before stating my ...
- 161