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Results tagged with symmetric-polynomials
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user 1056
2
votes
On a positivity property of Hall-Littlewood polynomials
There is a nice formula for Hall-Littlewood polynomials that follows from the combinatorial formula due to Haglund, Haiman and Loehr:
In this paper, eqution (69), we have that the Macdonald P-polyno …
- 15.8k
1
vote
On a positivity property of Hall-Littlewood polynomials
The specialization $P_\lambda(x;-1)$ is what is referred to as Schur's P functions. They can be described combinatorially using shifted tableaux,
and are Schur-positive.
See also slides here by S. Ch …
- 15.8k
2
votes
Polynomial inequality $n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3$
This is just a long comment, but translating to the notation of symmetric functions, you ask if
whenever $e_{111}(x) \geq 0$ and $e_3(x) \geq 0$, we have
$$
n^2 p_{(3)}(x) \geq p_{111}(x).
$$
This lat …
- 15.8k
0
votes
sum of squares of Schur polynomials indexed over partition valued functions on a set
Note that $s_\lambda(1^d)$ can be computed explicitly via Weyls formula on wikipedia in terms of the parts of $\lambda$.
- 15.8k
0
votes
Schur positivity of a polynomial
A similar case which I know about are the Boolean symmetric functions,
introduced by L. Billera, S. Billey, and V. Tewari.
They prove Schur positivity by using Chern plethysm, which I do not know much …
- 15.8k
1
vote
Expansion of key polynomials in terms of non-symmetric Hall-Littlewood polynomials and charg...
I have thought about this problem (well, a very similar one) a lot, and have some unpublished notes. There are several very annoying observations that seems to make defining charge very difficult. I t …
- 15.8k