Questions tagged [schur-functions]
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37
questions with no upvoted or accepted answers
18
votes
0
answers
375
views
Deforming a basis of a polynomial ring
The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions $\{S_\...
16
votes
0
answers
545
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 ...
15
votes
0
answers
244
views
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, ...
13
votes
0
answers
1k
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Generalization of Cauchy's identity
Let $ s_{\lambda} $ be the Schur function associated to the partition $ \lambda $.
Cauchy's identity (as in Macdonald) states that
$$
\sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = \prod_{i,j}(1-...
8
votes
0
answers
641
views
Cut-and-join equation and Schur function identity
This is somewhat related to my last MO post:
sum of the character of the symmetric group
Let $p_n$ be the $n$-th Newton symmetric function, and $s_{\nu}$ be the Schur function indexed by the ...
6
votes
0
answers
229
views
Derivations for symmetric functions
A symmetric function is a formal power series in infinitely many variables $x_1,x_2,\dots$ invariant under the permutation of variables (as opposed to a polynomial). Let $\Lambda$ denote the algebra ...
6
votes
0
answers
254
views
Macdonald's "Symmetric Functions and Hall Polynomials" Section 1.5 Example 9
I'm trying to follow Example 9 in Section 1.5 of the 2nd edition of Macdonald's book "Symmetric Functions and Hall Polynomials". I have trouble with understanding some points.
Before stating my ...
6
votes
0
answers
254
views
a variational problem related to weighted logarithmic capacity
Consider the following multiple contour integral:
$$ \Phi_\lambda := \oint \ldots \oint \prod_{1 \le j < k \le n} (z_j^{-1} - z_k^{-1}) \prod_{j=1}^n \prod_{k=1}^n (1 - z_j x_k)^{-1} \prod_{j=1}^n ...
5
votes
0
answers
117
views
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) \, =
\,
\...
5
votes
0
answers
396
views
Staircase Schur functions squared
Let $\Delta_n$ be the staircase-shaped partition $(n-1,n-2,\dots,1)$. Are there any non-obvious combinatorial objects that index $s_{\Delta_n}^2$? Here, $s_\lambda$ is the Schur function indexed by ...
4
votes
0
answers
96
views
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
votes
0
answers
112
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 $\...
4
votes
0
answers
201
views
Shifted Schur functions
Let's fix the ground field $\mathbb{C}$.
In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 Andrei Okounkov and Grigori Olshanski introduce a special basis for the center $Z(\...
4
votes
0
answers
202
views
Optimization problem involving Multivariate Normal
I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
$$h(\mu_{1},\ldots,\...
4
votes
0
answers
241
views
Analogy between canonical basis of U(n_-) and Schur functors, each under restriction
.1. For any category $\mathcal C$, possibly enriched over schemes, define $Rep({\mathcal C})$ to be the functor category ${\mathcal C} \to {\bf Vec}$ with direct sum inherited from $\bf Vec$. (If $\...
3
votes
0
answers
160
views
Factorization of symmetric polynomials
Let $\Lambda_n$ be the algebra of all symmetric polynomials in $n$ variables, which we also consider as an infinite-dimensional vector $\mathbb{Q}$-space, whose basis is the Schur polynomials.
The ...
3
votes
0
answers
105
views
"Sufficiently fat" Schur polynomial times Schubert polynomial is Schubert-positive
It's still open to find a combinatorial proof that for integers $k>0$ and $m>0$, a permutation $u\in S_\infty$ with its final descent at position $m$, and a partition $\lambda$ that
$$\mathfrak{...
3
votes
0
answers
133
views
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 ...
3
votes
0
answers
119
views
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 ...
3
votes
0
answers
242
views
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 &= \...
3
votes
0
answers
243
views
Dimension of roots of irreducible Schur polynomial on unit circle
Let $s_\lambda(x_1,\ldots,x_n)$ be a Schur polynomial in $\mathbb{C}[x_1,\ldots,x_n]$ with $\lambda=(\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n=0)$ and $\gcd(\lambda_1+n-1,\lambda_2+n-2,\dots,\...
3
votes
0
answers
720
views
Proofs that the Plücker relations generate the ideal of the Grassmannian
Some context: The $(k,n)$-Grassmannian is the set of $k$-dimensional subspaces of an $n$-dimensional vector space $V$. It can be realized as a projective variety via the Plücker embedding, and the set ...
3
votes
0
answers
324
views
what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions.
I am trying to understand first how one can define the plethysm say $s_\lambda \circ s_\mu$ as a module in the regular representation of the symmetric group.
1)How is it connected to the plethysms ...
2
votes
0
answers
90
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:...
2
votes
0
answers
112
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 ...
2
votes
0
answers
66
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 $w_1, w_2,\ldots$.
$$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\...
2
votes
0
answers
84
views
Schur function on unit circles
Define $T^d$ as following
$$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{C}^{d}\mid |t_i|= 1 \mbox{ for all } i\right\}
$$
For any partition $\lambda\vdash n$,The Schur function is defined
$$
\...
2
votes
0
answers
103
views
Bounding Schur polynomials of a particular shape
Consider Schur polynomials $s_\lambda$ with $\lambda = (2m, m, m, \ldots, m, 0)$ and $\ell(\lambda) = n$ (that is, $\lambda$ has $n$ rows). Here $m \gg n$, which, for the sake of concreteness, let's ...
2
votes
0
answers
2k
views
Every antisymmetric (alternating) polynomial is divisible by Vandermonde product
I am looking for a proof of the statement:
Every antisymmetric (alternating) polynomial is divisible by Vandermonde product, yielding a symmetric polynomial in the result.
The statement is really ...
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 ...
1
vote
0
answers
246
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 ...
1
vote
0
answers
29
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 $\lambda =[d]$ ...
1
vote
0
answers
214
views
Explicit basis/weight vectors for irreducibles inside the plethysm $Sym^m(\bigwedge^p \mathbf(V))$
This is a follow up to this question about finding the multiplicities of irreducible representations restricted to Young diagrams of 2-columns or less, inside the plethysm $Sym^m(\bigwedge^p \mathbf(V)...
0
votes
0
answers
123
views
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
votes
0
answers
105
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
0
answers
92
views
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 $\mathbb{S}_{\rho_1; \rho_2}$. ...
0
votes
0
answers
148
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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 $b_n$, $...