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edited Sep 10, 2018 at 23:14

In study of the cohomology ring of the Grassmannians, which is usually known as Schubert calculus, one usually deals with a distinguished basis known as the Schubert basis $\{\sigma_\lambda\}$. One of the most properties of this basis is positivity, the fact that for any two basis elements $\sigma_\lambda$ and $\sigma_\nu$, the multiplication constants $$ \sigma_\lambda \bullet \sigma_\mu = \sum_{\nu} c_{\lambda,\mu}^{\nu} \sigma_{\nu} $$ satisfy the positivity condition $$ c_{\lambda,\mu}^{\nu} > 0, ~~~~~~~~ \text{ for all } \nu. $$$$ c_{\lambda,\mu}^{\nu} \geq 0, ~~~~~~~~ \text{ for all } \nu. $$ Searching the literature, there seem to be a number of different proofs of this property, the relation between which is not always clear. Can people out there offer an opinion on which is the most insightful approach to proving positivity and what are the advantages/disadvantages or intuitons offered by the other approaches.

In study of the cohomology ring of the Grassmannians, which is usually known as Schubert calculus, one usually deals with a distinguished basis known as the Schubert basis $\{\sigma_\lambda\}$. One of the most properties of this basis is positivity, the fact that for any two basis elements $\sigma_\lambda$ and $\sigma_\nu$, the multiplication constants $$ \sigma_\lambda \bullet \sigma_\mu = \sum_{\nu} c_{\lambda,\mu}^{\nu} \sigma_{\nu} $$ satisfy the positivity condition $$ c_{\lambda,\mu}^{\nu} > 0, ~~~~~~~~ \text{ for all } \nu. $$ Searching the literature, there seem to be a number of different proofs of this property, the relation between which is not always clear. Can people out there offer an opinion on which is the most insightful approach to proving positivity and what are the advantages/disadvantages or intuitons offered by the other approaches.

In study of the cohomology ring of the Grassmannians, which is usually known as Schubert calculus, one usually deals with a distinguished basis known as the Schubert basis $\{\sigma_\lambda\}$. One of the most properties of this basis is positivity, the fact that for any two basis elements $\sigma_\lambda$ and $\sigma_\nu$, the multiplication constants $$ \sigma_\lambda \bullet \sigma_\mu = \sum_{\nu} c_{\lambda,\mu}^{\nu} \sigma_{\nu} $$ satisfy the positivity condition $$ c_{\lambda,\mu}^{\nu} \geq 0, ~~~~~~~~ \text{ for all } \nu. $$ Searching the literature, there seem to be a number of different proofs of this property, the relation between which is not always clear. Can people out there offer an opinion on which is the most insightful approach to proving positivity and what are the advantages/disadvantages or intuitons offered by the other approaches.

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asked Sep 10, 2018 at 22:39

Pierre Dubois

Proving Positivity for Schubert Calculus

In study of the cohomology ring of the Grassmannians, which is usually known as Schubert calculus, one usually deals with a distinguished basis known as the Schubert basis $\{\sigma_\lambda\}$. One of the most properties of this basis is positivity, the fact that for any two basis elements $\sigma_\lambda$ and $\sigma_\nu$, the multiplication constants $$ \sigma_\lambda \bullet \sigma_\mu = \sum_{\nu} c_{\lambda,\mu}^{\nu} \sigma_{\nu} $$ satisfy the positivity condition $$ c_{\lambda,\mu}^{\nu} > 0, ~~~~~~~~ \text{ for all } \nu. $$ Searching the literature, there seem to be a number of different proofs of this property, the relation between which is not always clear. Can people out there offer an opinion on which is the most insightful approach to proving positivity and what are the advantages/disadvantages or intuitons offered by the other approaches.