Interactive tool for enumerating Gelfand-Tsetlin schemes and displaying them as semi-standard Young tableaux. Computes Schur polynomials with optional checkerboard weight parameter a.
(≤ N rows; skew shape λ/μ)
GT schemes:0 | Polynomial terms:0 |
a-Kostka polynomials Kλ,μ(a)
Branching rule (M=1)
Statement. Define the checkerboard weight: for an SSYT $T$ of shape $\lambda/\mu$,
$$\mathrm{wt}_a(T) = \prod_{i} x_i^{m_i(T)} \cdot a^{\#\{(r,c) \in T : r+c \text{ even},\; \text{entry odd}\}}$$
where $m_i(T)$ is the number of entries equal to $i$. Then
$$s_{\lambda}^{(a)}(x_1,\ldots,x_N) = \sum_{\mu:\,\lambda/\mu\text{ horiz.\ strip}} s_{\mu}^{(a)}(x_1,\ldots,x_{N-1})\cdot x_N^{|\lambda/\mu|}\cdot a^{d(\lambda/\mu)}$$
where $d(\lambda/\mu) = \#\{(r,c)\in \lambda/\mu : r+c\text{ even}\}$ if $N$ is odd, and $0$ if $N$ is even.
Mathematica code
code
(note: parameters in the code might differ from the ones in
simulation results below)
Link to code
(GT schemes, SSYT enumeration, and Schur polynomials with checkerboard weight)
Dear colleagues:
Feel free to use code (unless otherwise specified next to the corresponding link),
data, and visualizations to illustrate your research in talks and papers,
with attribution (CC BY-SA 4.0 (opens in new tab)).
Some images are available in very high resolution upon request.
I can also produce other simulations upon request - email me at lenia.petrov@gmail.com
This material is based upon work supported by the National Science Foundation under Grant DMS-2153869