Abstract
Recall the classical result that the cup product structure constants for the singular cohomology with integral coefficients $$H^*({{\mathrm{\mathrm {Gr}}}}(r, n))$$ of the Grassmannian of r-planes coincide with the Littlewood-Richardson tensor product structure constants for $${{\mathrm{\mathrm {GL}}}}_r$$ . Specifically, the result asserts that there is an explicit surjective ring homomorphism $$\xi : {{\mathrm{\mathrm {Rep}}}}_{{{\mathrm{\mathrm {poly}}}}}({{\mathrm{\mathrm {GL}}}}_r) \rightarrow H^*({{\mathrm{\mathrm {Gr}}}}(r, n))$$ , where $${{\mathrm{\mathrm {Gr}}}}(r, n)$$ denotes the Grassmannian of r-planes in $$\mathbb {C}^n$$ and $${{\mathrm{\mathrm {Rep}}}}_{{{\mathrm{\mathrm {poly}}}}} ({{\mathrm{\mathrm {GL}}}}_r)$$ denotes the polynomial representation ring of $${{\mathrm{\mathrm {GL}}}}_r$$ . This work seeks to achieve one possible generalization of this classical result for $${{\mathrm{\mathrm {GL}}}}_r$$ and the Grassmannian $${{\mathrm{\mathrm {Gr}}}}(r,n)$$ to the Levi subgroups of any reductive group G and the corresponding flag varieties.
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