Questions tagged [schur-functions]
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88
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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).
...
- 81
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votes
0
answers
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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
$$\mathfrak{...
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Representation-theoretic interpretation of double Schur polynomials
The Schur polynomials
$$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}}$$
naturally appear as polynomial representatives for Schubert classes in ...
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1
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Is this simple symmetry of Littlewood-Richardson coefficients known?
Let $\lambda$ be a partition with at most $p$ parts, let $\mu$ be a partition with at most $q$ parts, and let $\nu$ be a partition with at most $p+q$ parts. Let $m\geq \nu_1$ be an integer. We denote ...
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votes
1
answer
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The convex hull of Schur polynomial evaluations
Let $r\leq n$ and $d$ be positive integers. A probability vector is a vector of non-negative entries that sum to 1. For each probability vector $\lambda$ of length $n$, let
$$s(\lambda)=(\dim[\pi] \...
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votes
2
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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 ...
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Is it express in terms of Schur Q-function?
Consider next integral
\begin{eqnarray}
Z \ = \ h^{- N N_f} \ \int\limits_{SU(N)} \ dU \ \prod_{n=1}^{N} \
\det \left ( 1 + h U \right )^{ N_f} \
\left ( 1 + h U^{\dagger} \right )^{ N_f} \ = \sum_{...
0
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0
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views
I search representation in terms of Schur Q-function
Consider next sum
$$
Z_0^{N, N_f} = \sum_{r=0}^{N N_f} \sum_{\lambda \vdash r }s_{\lambda}(1^{N_f})
s_{\lambda} (1^{N_f})
= \det_{1\le i, j \le N} \ \binom{2 N_f}{N_f-i +j} = s_{N^{N_f}} \left(...
0
votes
1
answer
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Dose density matrix with off-diagonal elements equal to zero has maximum von-Neumann entropy?
von-Neumann entropy
I know von-Neumann entropy on density matrix $S=-{\rm Tr}(\rho \ln\rho)$ is similar to Shannon entropy $S=-\sum_i p_i\ln p_i$ in classical mechanics. And I want to get Bose-...
- 3
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Sum of Schur functions associated to self-conjugate partitions
The $\tau$-function $H^\circ \big(t ;\vec{x} \big)$ associated with counting simple Hurwitz numbers is the formal power series
\begin{equation}
(\dagger) \quad H^\circ \big(t ;\vec{x} \big) \, =
\,
\...
- 1,797
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votes
0
answers
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Recovery from $p$-norms: Question on proof
I am reading a paper on recovery from power sums. A main statement of the paper is that for a vector $(a_1,...,a_n) \in \mathbb{N}^n$ with pairwise distinct $a_i$ and the map
$$\phi:\begin{cases}
...
- 83
2
votes
1
answer
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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 ...
- 15.4k
8
votes
1
answer
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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(\...
- 1,797
4
votes
1
answer
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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 ...
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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 ...
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votes
1
answer
403
views
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
$$
D_\lambda(f)...
- 516
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votes
1
answer
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Generating function for Schur polynomials
Consider the generating function
$$
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},
$$
where the sum ...
- 516
2
votes
1
answer
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How to re-expand the sum of Schur function?
Consider next sum
\begin{eqnarray}
\label{PF_spindef}
Z = \sum_{r=0}^{N N_f} h^{2r} \ Q(r) .
\end{eqnarray}
and
\begin{equation}
Q(r) \ = \ \sum_{\sigma \vdash r} s_{\sigma}(1^{N_f}) \
s_{\sigma}...
12
votes
2
answers
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Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule
The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...
- 431
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votes
2
answers
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About the sum of rectangular power sums
Let $n \geq 1$ be an integer and consider the symmetric function
$$D_n = \sum_{d|n} p_d^{n/d},$$
where $p_{d}$ are the power-sum symmetric functions.
It can be checked up to $n=35$ that the symmetric ...
- 3,467
1
vote
0
answers
68
views
LGV scheme: Any situations where the weights shift differently for each path?
In Cylindric partitions, Proposition 1, Gessel and Krattenthaler prove a formula for lattice paths on a cylinder
In our particular problem, we again have paths $((P_{1},k_{1}),...,(P_{r},k_{r}))$ but ...
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Tensors of minimal rank in Schur modules $S_{\lambda}V \subset V^{\otimes |\lambda|}$
It is well known that for a vector space $V$ with $\dim(V)=n+1$ the $GL(V)-$module $V^{\otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{\lambda}V$, where $\...
- 1,303
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proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions"
I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...
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Double Schur function expansion
In literature, I have seen the weighted Hurwitz number $N_{g,n}(d_1 , d_2 \ldots , d_n)$ which are symmetric number and they can be written as double Schur function expansion.
\begin{align} \label{eq:...
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LGV scheme for lattice paths that move in non-unit spatial positive steps
In the Lindström–Gessel–Viennot lemma (LGV) applied to the $Z^2$-lattice paths are taken to move in unit spatial-steps in unit time (see here).
What do we mean by "time"? In the language of ...
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votes
2
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Do you know an elegant proof for this expression for a Schur function?
I know that the identity
$$
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}
$$
holds.
Here $\alpha=1^{...
- 5,504
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Derivations for symmetric functions
A symmetric function is a formal power series in infinitely many variables $x_1,x_2,\dots$ invariant under the permutation of variables (as opposed to a polynomial). Let $\Lambda$ denote the algebra ...
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Theorem 5.3 ([Okounkov-01]) in Borodin and Gorin's lecture note
In this lecture note: https://arxiv.org/pdf/1212.3351.pdf, Theorem 5.3(P28):
Suppose that the $\lambda \in \mathbb{Y}$ is distributed according to the Schur measure $\mathbb{S}_{\rho_1; \rho_2}$. ...
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What is an orthogonal form?
I am reading the article: Algebro - Geometric applications of Schur S- and Q-polynomials
On page 179, they said about an orthogonal form $\psi$ on $W$. There is no definition of an orthogonal form on ...
- 320
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vote
1
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Schur polynomials with zeros in an infinite geometric progression
Let $a_1,...,a_n\in \overline{\mathbb{Q}}^\times$ be algebraic numbers, such that $\frac{a_i}{a_j}$ is not a root of unity for $i\neq j$. Furthermore, let $m\in\mathbb{N}$, $m>1$.
Question: Is ...
- 171
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votes
1
answer
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Finding Littlewood-Richardson coefficients without using identities
The Littlewood-Richardson coefficients $C^{R}_{QP}$ for some partitions $R, Q, P$ can usually be dealt with using identities like for example $$C^{R}_{QP} = 0 \quad \text{ if } \quad|R| \neq |Q| + |P|...
1
vote
0
answers
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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
$$...
3
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0
answers
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Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)
For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...
- 131
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Restricted Cauchy identity
Is there some reference for sums like:
$$\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|$$
(summation ...
- 625
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1
answer
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Decomposing tensor powers of the fundamental representation of exceptional Lie algebras
For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, ...
- 805
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Annihilator of the of the generating function not holonomic
The following is a generating function in $x,h$ with infinite parameters
$q_1,q_2\ldots,$ and $w_1, w_2,\ldots$.
$$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\...
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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 $\lambda =[d]$ ...
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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
$$
\...
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votes
1
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Is there a Giambelli identity with dual representations?
For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition.
...
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Integral of product of Schur functions
Schur functions are irreducible characters of the unitary group $\mathcal{U}(N)$. This implies the integration formulae
$$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^...
- 2,460
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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 ...
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Shifted schur function and holonomic
Now let us denote by $\Lambda^{*}(n)$ the algebra of polynomials in $x_{1},\ldots,x_{n}$ that become symmetric in new variables
$$ x_{i}'=x_{i}-i+c, \ i \in 1,\ldots,n.$$
Here c is a arbitrary fixed ...
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votes
0
answers
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Generalization of Newton's identities to Schur functions
In some recent work, I've stumbled across the following identity for $\lambda \vdash n$:
$$
n s_\lambda = \sum_{k=1}^n p_k \sum_{\mu \nearrow_k \lambda} (-1)^{\mathrm{ht}(\lambda/\mu)} s_\mu.
$$
Here, ...
- 1,849
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votes
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Frobenius coordinate expansion of character
Let $\lambda$ be the partition of integer $d$. The Frobenius coordinate of $\lambda$ is given
$$ (a_1 ,\ldots,a_{d(\lambda)}|b_1,\ldots,b_{d(\lambda)}),$$
where $d(\lambda)$ denote the diagonal of $\...
- 675
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votes
0
answers
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Evaluating derivatives of Schur polynomials
Given an arbitrary partition $\lambda$ and an integer $N$ (the number of variables), is there any further way to evaluate the following derivative of the Schur polynomial?
\begin{align}
A &= \...
2
votes
1
answer
1k
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An upper bound for a vector with given norm 1 and norm 2
Suppose $X = (x_1, \ldots , x_n)$ is given and we know that $x_i$'s are nonnegative, $\sum_{i=1}^n x_i = n$ and $\sum_{i=1}^n x_i^2 = m $. Just by this information, is it possible to find a vector ...
- 228
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Methods to get Holonomic functions
Let $a_n$ be a holonomic sequence. By definition, that means there exists a linear differential equation of finite order which annihilates $F(x)$, where
$F(x):=\sum a_n x^n$.
Similarly let $b_n$, $...
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votes
1
answer
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Decompostion of hook schur function in terms of cauchy product of holonomic functions
Let $s_{\lambda}$ denote the schur function and $\lambda$ is the partion of an integer. The schur function written in power sum symmetric basis apper as following. $\chi$ denote the character.
\begin{...
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votes
1
answer
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Is there a geometric interpretation of skew Schur functions?
Consider the cohomology ring of the Grassmannian of k-planes in complex n-space. It has a standard presentation as a quotient of the ring of symmetric functions. In this presentation, the Schur ...
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Why do we care about Schur Positivity
Some of the most important open problems in Algebraic Combinatorics concern the Schur positivity of classes of symmetric functions. Why is this an important property to have?
