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?
> 
> 
> 



Weyl map for $SU(n)$
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
$$
p^\* x\_{2i-1} = \sum\_{k=1}^n\Big[\sum\_{\substack{S\subset \{1,\dots,n\}\smallsetminus\{k\}\\|S| = i - 1}}\prod\_{l\in S} y\_l\Big]\otimes z\_k\ .
$$


