All Questions
Tagged with algebraic-combinatorics symmetric-functions
12
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 ...
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
1
vote
1
answer
121
views
Cycle indicator symmetric function and Polya's theorem
Let $G$ be a subgroup of the symmetric group $S_n$. Given a partition $\lambda$ of $n$, denote by $n_G(\lambda)$ the number of elements in $G$ of cycle-type $\lambda$. The cycle indicator function of $...
- 3,052
3
votes
0
answers
106
views
skew Schur identity
Let $\lambda$ be a partition of size at least 2, and let $n>0$ be an integer. Prove that
$$s_{\lambda/(1)}.h_{n}=\sum_{\lambda^{+}\supseteq \lambda\\\lambda^{+}/\lambda\ \text{hor.} \ n\ \text{...
- 991
0
votes
1
answer
48
views
reference request: Type A crystal proof of Schur-positivity
In this question: https://mathoverflow.net/questions/272306/what-techniques-are-there-to-prove-schur-positivity, one of the techniques listed to prove Schur-positivity is called a Type-A crystal proof,...
- 841
8
votes
2
answers
381
views
Proving an identity for complete homogenous symmetric polynomials
Probably everybody knows the expression:
$$
\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},
$$
where $a_1\ne a_2$ is assumed.
It seems that it can be further ...
- 26.7k
4
votes
2
answers
581
views
What do we mean when we say the Schur functions form a basis.
This has always bugged me. When we are examining symmetric functions (or polynomials if you prefer finitely many variables), we have an easy choice of basis with the monomial symmetric functions. As ...
2
votes
1
answer
1k
views
nth power symmetric polynomial in terms of Schurs polynomial
The Schur's polynomial forms the basis of the symmetric algebra so does the power symmetric function. nth power symmetric function are the function of the form $\sum_i x_i^n$. Let $\lambda \vdash n$ ...
- 1,065
1
vote
1
answer
169
views
prove a polynomial identity..
The equation is that
$h_m(x_1, \cdots, x_n, a)-h_m(x_1, \cdots, x_n, b)=(a-b)h_{m-1}(x_1, \cdots, x_n, a, b)$ where $h_m$ is a complete homogeneous symmetric polynomial.
See and find several ...
- 313
4
votes
1
answer
104
views
Can the natural proof of this algebraic identity be simplified?
Let $x^4+c_3x^3+c_2x^2+c_1x+c_0$ be a real polynomial with no real root. Then there are two pairs
of conjugate complex roots, $a_1\pm b_1 i$ and $a_2\pm b_2 i$, and one has the identity
$$
c_1^2-...
- 61.7k
6
votes
2
answers
992
views
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
1
answer
114
views
Prove that $h_r(x_1,\dots,x_n)=\sum^n_{k=1}x^{n-1+r}_k\prod_{i\neq k}(x_k-x_i)^{-1}$
Let $h_r\left(x_1,\ldots,x_n\right)$ denote the $r$-th complete homogeneous symmetric polynomial in the $n$ indeterminates $x_1, \ldots, x_n$ (that is, the sum of all degree-$r$ monomials in these $n$ ...
- 31