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Let $G= SU(n)$ and let $\mathbb{T}$ be the maximal torus in $G$ given by diagonal matrices. We have $$ H^*(G,\mathbb{Q}) \cong \Lambda_{\mathbb{Q}}[x_3, x_5, \dots, x_{2n-1}] \ . $$ Now consider the Weyl map $$ p \colon G/\mathbb{T} \times \mathbb{T} \to G \quad , \quad ([g],z) \mapsto gzg^{-1}\ . $$ The induced map in rational cohomology $$ p^* \colon H^*(G,\mathbb{Q}) \to H^*(G/\mathbb{T}, \mathbb{Q}) \otimes H^*(\mathbb{T},\mathbb{Q}) $$ is injective. In fact, if we restrict the codomain to fixed points with respect to a certain action of the Weyl group $W$ it becomes an isomorphism (see for example Reeder, On the Cohomology of Compact Lie Groups). The cohomology ring $H^*(\mathbb{T},\mathbb{Q})$ is isomorphic to another exterior algebra and there are also explicit descriptions of $H^*(G/\mathbb{T},\mathbb{Q})$ (see Reeder's paper again for a reference).

Is there a formula describing $p^*(x_{2i+1})$ in terms of any set of natural generators for the codomain?

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First, let me fix generators for $H^*(SU(n))$ and $H^*(SU(n)/\mathbb T)$: For the first, consider the vector bundle on $\Sigma SU(n)$ with clutching map $\operatorname{id}_{SU(n)}$, i.e. with classifying map $f_n\colon\Sigma SU(n)\simeq \Sigma\Omega BSU(n)\to BSU(n)$, and let $\Sigma x_{2i-1} = f^*c_i$. For the second, let $\pi_k\colon \mathbb T\subset U(1)^k\to U(1)$ be the projection to the $k$-th coordinate and $L_k = SU(n)\times_{T,\pi_k} \mathbb C$ be the associated line bundle, and set $y_k = c_1(L_k)$; since $\bigoplus_{k=1}^n \pi_k$ is the restriction of the defining representation of $SU(n)$ to $\mathbb T$, the sum $E := \bigoplus_{k=1}^n L_k$ is trivial, so that all symmetric polynomials in the $y_k$ vanish since they can be expressed via Chern classes of $E$. The induced map $Q[y_1,\dots,y_n]/Q[y_1,\dots,y_n]^{S_n}\to H^*(SU(n)/\mathbb T)$ is an isomorphism; a basis for this cokernel is given by the monomials $y_1^{\alpha_1}\dots y_n^{\alpha_n}$ with $0\le \alpha_k < k$.

Now think of $p:G/\mathbb T\times\mathbb T\to SU(n)$ as an automorphism $\phi$ of $E\boxtimes\underline{\mathbb C}$; it splits as $\phi = \phi_1\oplus\dots\oplus \phi_n$, where $\phi_k = \operatorname{id}_{L_k}\boxtimes \pi_k$ is the automorphism of $L_k\boxtimes \underline{\mathbb C}$ obtained as the external tensor product of the identity of $L_k$ with the projection $\pi_k:\mathbb T\subset U(1)^k\to U(1)$.

Consider the clutching construction $(E\boxtimes\underline{\mathbb C})^\phi\to G/\mathbb T\times\mathbb T\times S^1$ which is obtained by gluing the two ends of $E\boxtimes\underline{\mathbb C}$ together using $\phi$. Denote by $z_k = \pi_k^*([U(1)]), u = [S^1]$ the obvious cohomology classes in degree $1$, and itdentify them and the $y_k$ with their image in the product. An easy argument shows that $c_1(\mathbb C^{\pi_k}) = uz_k\in H^2(\mathbb T\times S^1)$. By naturality, we have \begin{align*} (E\boxtimes\underline{\mathbb C})^\phi &\cong \bigoplus_{k=1}^n(L_k\boxtimes\underline{\mathbb C})^{\phi_k}\\ c_1\big((L_k\boxtimes\underline{\mathbb C})^{\phi_k}\big) &= c_1(L_k) + c_1(\underline{\mathbb C}^{\pi_k}) = y_k + uz_k\\ c\big((E\boxtimes\underline{\mathbb C})^\phi\big) &= \prod_{k=1}^n c\big((L_k\boxtimes\underline{\mathbb C})^{\phi_k}\big)\\ &= \prod_{k=1}^n (1 + y_k + uz_k)\\ &= \underbrace{c(E)}_{= 0} + u\sum_{k=1}^n z_k \sum_{S\subset \{1,\dots,n\}\smallsetminus\{k\}}\prod_{l\in S} y_l \end{align*}

It is no surprise that this expression vanishes for $u=0$ since we can use the chosen trivialization of $E\boxtimes \underline{\mathbb C}$ to descend $(E\boxtimes \underline{\mathbb C})^\phi$ to $\Sigma(G/\mathbb T\times \mathbb T)$, and the resulting vector bundle has classifying map $f_n\circ \Sigma p$. Chasing through the definitions, we see that $$ p^* x_{2i-1} = \sum_{k=1}^n\Big[\sum_{\substack{S\subset \{1,\dots,n\}\smallsetminus\{k\}\\|S| = i - 1}}\prod_{l\in S} y_l\Big]\otimes z_k\ . $$

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