All Questions
Tagged with algebraic-combinatorics symmetric-polynomials
13
questions
2
votes
1
answer
51
views
Regarding the number of variables in Symmetric Functions
I'm studying Symmetric Functions and I came across a doubt that could be considered stupid but I need clarifications.
In the course I'm following we introduced symmetric functions as formal series of ...
0
votes
0
answers
14
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Isometric automorphisms of the ring of symmetric functions
I was trying to understand how special the $\omega$ involution on the ring of symmetric functions $\Lambda$ or $\Lambda^n$ (restriction to $n$ variables, just in case if by some magic, the situation ...
- 143
2
votes
0
answers
22
views
Derivation of the Macdonald operator $D_{n}(X;q,t)$
Since I first encountered Equation (3.2) on p.315 of Macdonald's Symmetric functions and Hall polynomials, I have wanted to know where it comes from. So how does one derive the operator
\begin{...
- 98
4
votes
1
answer
64
views
Can the product of two quasisymmetric functions (that are not symmetric) be symmetric?
Given two quasisymmetric functions (see definition below) that are not symmetric, must their product also not be symmetric?
From Stanley's Enumerative Combinatorics 2, recall that a function $f\in \...
- 83
0
votes
0
answers
107
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Hall Scalar product calculation: $\langle h_3p_6, s_{(5,4)}\rangle$
This is a problem from an old qualifying exam I am reviewing:
Calculate: $\langle h_3p_6, s_{(5,4)}\rangle$ where $\langle \cdot,\cdot \rangle$ is the Hall scalar product defined by $\langle s_\lambda,...
1
vote
1
answer
173
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Help with an expression of the Schur polynomial
I am reading: https://www2.math.upenn.edu/~peal/polynomials/schur.htm where it says:
The Schur polynomial is defined as $$s_\lambda(x_1,...,x_n)=\prod_{1\le i<j\le n}(x_i-x_j)^{-1}\det(x_j^{\...
- 5,760
8
votes
2
answers
382
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Proving an identity for complete homogenous symmetric polynomials
Probably everybody knows the expression:
$$
\sum_{k_1,k_2\ge0}^{k_1+k_2=k}a_1^{k_1}a_2^{k_2}=\frac{a_1^{k+1}-a_2^{k+1}}{a_1-a_2},
$$
where $a_1\ne a_2$ is assumed.
It seems that it can be further ...
- 26.7k
1
vote
0
answers
96
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Murnaghan-Nakayama rule for general dimension of a hook
Let $m,r\in \mathbb{N}, n=rm$. The character of symmetric group $\chi^{\lambda}$ where $\lambda$ is of the from
$(k,1^{n-k})$ evaluated at the conjugacy class of $S_{n}$ of the from $(r,\ldots,r)$, ...
- 1,065
4
votes
2
answers
581
views
What do we mean when we say the Schur functions form a basis.
This has always bugged me. When we are examining symmetric functions (or polynomials if you prefer finitely many variables), we have an easy choice of basis with the monomial symmetric functions. As ...
2
votes
1
answer
1k
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nth power symmetric polynomial in terms of Schurs polynomial
The Schur's polynomial forms the basis of the symmetric algebra so does the power symmetric function. nth power symmetric function are the function of the form $\sum_i x_i^n$. Let $\lambda \vdash n$ ...
- 1,065
1
vote
1
answer
169
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prove a polynomial identity..
The equation is that
$h_m(x_1, \cdots, x_n, a)-h_m(x_1, \cdots, x_n, b)=(a-b)h_{m-1}(x_1, \cdots, x_n, a, b)$ where $h_m$ is a complete homogeneous symmetric polynomial.
See and find several ...
- 313
6
votes
2
answers
992
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Properties of the 'forgotten' symmetric polynomials
In I.G. Macdonald "Symmetric Functions and Hall Polynomials" pg.22, the forgotten symmetric functions $f$ are introduced very briefly as the result of applying an involution $\omega$ to the monomial ...
- 910
3
votes
0
answers
234
views
Symmetrization of Powersum polynomials
Let $n\in\mathbb{N}$. Then, for $i\in\mathbb{N}$, the $i-$th power sum if defined to be the polynomial $p_i^{(n)}:=\sum_{j=1}^n x_j^i$ in $n$ indeterminates $x_1,x_2,\ldots,x_n$.
Then let $\lambda:=(\...
- 51