%0 Generic %1 kumar2022representation %A Kumar, Shrawan %A Xie, Jiale %D 2022 %K Representations %T Representation ring of Levi subgroups versus cohomology ring of flag varieties III %U http://arxiv.org/abs/2207.04537 %X For any reductive group $G$ and a parabolic subgroup $P$ with its Levi subgroup $L$, the first author Ku2 introduced a ring homomorphism $ \xi^P_łambda: Rep^C_łambda-poly(L) H^*(G/P, C)$, where $ Rep^C_łambda-poly(L)$ is a certain subring of the complexified representation ring of $L$ (depending upon the choice of an irreducible representation $V(łambda)$ of $G$ with highest weight $łambda$). In this paper we study this homomorphism for $G=SO(2n)$ and its maximal parabolic subgroups $P_n-k$ for any $2kn-1$ (with the choice of $V(łambda) $ to be the defining representation $V(ømega_1) $ in $C^2n$). Thus, we obtain a $C$-algebra homomorphism $ \xi_n,k^D: Rep^C_ømega_1-poly(SO(2k)) H^*(OG(n-k, 2n), C)$. We determine this homomorphism explicitly in the paper. We further analyze the behavior of $ \xi_n,k^D$ when $n$ tends to $ınfty$ keeping $k$ fixed and show that $ \xi_n,k$ becomes injective in the limit. We also determine explicitly (via some computer calculation) the homomorphism $ \xi^P_łambda$ for all the exceptional groups $G$ (with a specific `minimal' choice of $łambda$) and all their maximal parabolic subgroups except $E_8$.

@misc{kumar2022representation, abstract = {For any reductive group $G$ and a parabolic subgroup $P$ with its Levi subgroup $L$, the first author [Ku2] introduced a ring homomorphism $ \xi^P_\lambda: Rep^\mathbb{C}_{\lambda-poly}(L) \to H^*(G/P, \mathbb{C})$, where $ Rep^\mathbb{C}_{\lambda-poly}(L)$ is a certain subring of the complexified representation ring of $L$ (depending upon the choice of an irreducible representation $V(\lambda)$ of $G$ with highest weight $\lambda$). In this paper we study this homomorphism for $G=SO(2n)$ and its maximal parabolic subgroups $P_{n-k}$ for any $2\leq k\leq n-1$ (with the choice of $V(\lambda) $ to be the defining representation $V(\omega_1) $ in $\mathbb{C}^{2n}$). Thus, we obtain a $\mathbb{C}$-algebra homomorphism $ \xi_{n,k}^D: Rep^\mathbb{C}_{\omega_1-poly}(SO(2k)) \to H^*(OG(n-k, 2n), \mathbb{C})$. We determine this homomorphism explicitly in the paper. We further analyze the behavior of $ \xi_{n,k}^D$ when $n$ tends to $\infty$ keeping $k$ fixed and show that $ \xi_{n,k}$ becomes injective in the limit. We also determine explicitly (via some computer calculation) the homomorphism $ \xi^P_\lambda$ for all the exceptional groups $G$ (with a specific `minimal' choice of $\lambda$) and all their maximal parabolic subgroups except $E_8$.}, added-at = {2022-07-12T09:55:04.000+0200}, author = {Kumar, Shrawan and Xie, Jiale}, biburl = {https://www.bibsonomy.org/bibtex/2bdf4a952b18097157873be9920dba3b1/dragosf}, description = {Representation ring of Levi subgroups versus cohomology ring of flag varieties III}, interhash = {716809b40e53d2c50d8b9d9f0f3267d1}, intrahash = {bdf4a952b18097157873be9920dba3b1}, keywords = {Representations}, note = {cite arxiv:2207.04537Comment: 28 pages}, timestamp = {2022-07-12T09:55:04.000+0200}, title = {Representation ring of Levi subgroups versus cohomology ring of flag varieties III}, url = {http://arxiv.org/abs/2207.04537}, year = 2022 }