2009/10/05 by Nicolas Ressayre, Ressayre, Nicolas
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT
paper · pdf · doi:10.48550/arxiv.0910.0697
openalex publication_date 2009/10/05 · arxiv created 2009/11/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a complex semisimple simply connected algebraic group. Given two irreducible representations V1 and V2 of G, we are interested in some components of V1⊗ V2. Consider two geometric realizations of V1 and V2 using the Borel-Weil-Bott theorem. Namely, for i=1, 2, let \Lii be a G-linearized line bundle on G/B such that \rm Hqi(G/B,\Lii) is isomorphic to Vi. Assume that the cup product \rm Hq1(G/B,\Li1)⊗ \rm Hq2(G/B,\Li2)\longto \rm Hq1+q2(G/B,\Li1⊗\Li2) is non zero. Then, \rm Hq1+q2(G/B,\Li1⊗\Li2) is an irreducible component of V1⊗ V2; such a component is said to be \it cohomological. Solving a Dimitrov-Roth conjecture, we prove here that the cohomological components of V1⊗ V2 are exactly the PRV components of stable multiplicity one. Note that Dimitrov-Roth already obtained some particular cases. We also characterize these components in terms of the geometry of the Eigencone of G. Along the way, we prove that the structure coefficients of the Belkale-Kumar product on \rm H^*(G/B,\ZZ) in the Schubert basis are zero or one.