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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!

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  • $\begingroup$ 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? $\endgroup$ – abx Feb 7 at 11:26
  • $\begingroup$ 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 $\ell_1 = \ell_2$. $\endgroup$ – FreddyG Feb 7 at 12:10
  • $\begingroup$ 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. $\endgroup$ – abx Feb 7 at 12:25
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    $\begingroup$ 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). $\endgroup$ – abx Feb 7 at 14:18
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    $\begingroup$ Ok i understand. Thank you for your answer ! $\endgroup$ – FreddyG Feb 7 at 14:20

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