# Flag variety as monoid and Schubert calculus

The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation. When looking at flags in V on could consider the sum of two flags using the monoid operation on subspaces. I guess it is then possible to provide flags with a structure of monoid (algebraic monoid ?). How is reflected this monoid structure in terms of Schubert varities or Schubert classes or Schubert polynomials etc. ? Thank you!

• This "sum" is not well-defined. Take $\dim V=2$, two flags given by lines $\ell_1,\ell_2$ in $V$. What is the sum?
– abx
Feb 7, 2021 at 11:26
• As a sum, this would be the smallest linear subspace containing both $\ell_1$ and $\ell_2$ that is we would have the partial flag $0 \subset V$ if $\ell_1 \neq \ell_2$ or $0\subset \ell_1\subset V$ if $\ell_1 = \ell_2$. Feb 7, 2021 at 12:10
• But this is not in the same flag variety... Maybe you want to take the disjoint union of all flag varieties built on $V$? That doesn't sound too promising.
– abx
Feb 7, 2021 at 12:25
• The answer is no, because Schubert calculus deals with the variety of flags with a fixed type (:= dimension of the subspaces appearing in the flag).
– abx
Feb 7, 2021 at 14:18
• Ok i understand. Thank you for your answer ! Feb 7, 2021 at 14:20