Hall 123

Demazure crystal structure for flagged skew tableaux and flagged reverse plane partitions

Siddheswar Kundu

IMSc

Given a skew shape $ \lambda / \mu $ and a flag $\Phi$, we show that the flagged dual
stable Grothendieck polynomial $g_{\lambda/\mu}(X_\Phi)$ is a non-negative sum of key
polynomials. We prove this by showing that the set of all flagged reverse plane partitions of
shape $\lambda / \mu$ and flag $\Phi$ admits a Demazure crystal structure, which generalizes our
previous result, namely, the set of all flagged semi-standard tableaux of shape $\lambda / \mu$
and flag $\Phi$ is a disjoint union of Demazure crystals. We use this fact to give a tableau
model for the flagged skew Littlewood-Richardson coefficients $c_{\lambda, \, \mu/\gamma}
^{\,
u} (\Phi)$, which are a generalization of the usual Littlewood-Richardson coefficients. We
further give a hive model for the coefficients $c_{\lambda, \, \mu/\gamma} ^{\,
u} (\Phi)$.
Finally, we establish the saturation property of these coefficients, generalizing the results of
Knutson-Tao and Kushwaha-Raghavan-Viswanath.