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Representation ring of Levi subgroups versus cohomology ring of flag varieties

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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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Acknowledgments

I am grateful to Michele Vergne whose question led me to this work. This work was partially done during my visit to the University of Sydney, hospitality of which is gratefully acknowledged. This work was partially supported by the NSF Grant DMS-1501094.

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  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599–3250, USA

    Shrawan Kumar

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  1. Shrawan Kumar
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Correspondence to Shrawan Kumar.

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Kumar, S. Representation ring of Levi subgroups versus cohomology ring of flag varieties. Math. Ann. 366, 395–415 (2016). https://doi.org/10.1007/s00208-015-1325-6

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  • DOI: https://doi.org/10.1007/s00208-015-1325-6

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