#### 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.

Done