# Noncommutative cohomology of flag varieties

Consider the Grassmannian $Gr(n, N)$ of $n$-dimensional subspaces of $\mathbf C^N$. Its cohomology ring is isomorphic to $\mathbf C[x_1, \ldots, x_n, \bar x_1, \ldots \bar x_{N-n}]/I_{n,N}$, wherethe ideal is generated by homogeneous elements of

$$(1+x_1t+\ldots+ x_nt^n)(1+\bar x_1 t+\ldots +\bar x_{N-n}t^{N-n}) -1$$

This ring is isomorphic to the ring

$$\frac{\mathbf C[\varepsilon_1, \ldots, \varepsilon_n]}{(\text{h_i=0 for i>N-n})}$$

of elementary symmetric polynomials $\varepsilon_i$ in the $x_i$, modulo the complete homogeneous symmetric polynomials $h_i$ for $i>N-n$.

Ellis/Khovanov/Lauda now show similar results for odd polynomials $\mathbf C\langle x_1, \ldots, x_n\rangle/(\text{$x_ix_j=-x_jx_i$for$i\neq j$})$, see . One can define odd analogues of the elementary and complete homogeneous symmetric polynomials. Then one can recover the odd analogue of the above quotient as some odd cohomology, see .

The even case can be generalized for $n$-step flag varieties $\{0=F_0\subseteq F_1 \subseteq\ldots\subseteq F_n = \mathbf C^N\}$, see [3, §5]. The respective ideal to be quotiented out then is generated by homogeneous parts of

$$(1+x_{11}t+x_{12}t^2+\ldots)(1+x_{21}t+x_{22}t^2+\ldots)\cdots -1,$$

and the “dual” elements $\bar x_{ij}$ are given by homogeneous parts of that ideal, leaving out the $i$-th factor.

Question: Can the results for odd symmetric polynomials and their relations to cohomology of Grassmannians be generalized to some odd cohomology of $n$-step flag varieties? I am struggling with the problem that I cannot pull out factors of the above ideal easily anymore in the non-commutative case.