Schubert class in the Grassmannian G(3,6)

Question

How to compute the Schubert class $\sigma$$^2$$_2$$_1$ in the Grassmannian G(3,6)?

I remember the result is $\sigma$$_3$$_3$ + 2$\sigma$$_3$$_2$$_1$ + $\sigma$$_2$$_2$$_2$.

Answer 1

Note that $\sigma_{2,1} = \sigma_2\cdot\sigma_1 - \sigma_3$, hence $$ \sigma_{2,1}^2 = \sigma_{2,1}\cdot\sigma_2\cdot\sigma_1 - \sigma_{2,1}\cdot\sigma_3. $$ By Pieri rule $$ \sigma_{2,1}\cdot\sigma_2\cdot\sigma_1 = (\sigma_{3,2} + \sigma_{3,1,1} + \sigma_{2,2,1})\cdot\sigma_1 = \sigma_{3,3} + 3\sigma_{3,2,1}+\sigma_{2,2,2}, $$ while $$ \sigma_{2,1}\cdot\sigma_3 = \sigma_{3,2,1}. $$ Subtracting, you get the result.

Alternatively, one can directly use the Littlewood-Richardson rule.