# Intersection of components in Springer fibre of type A

From the standard results on Springer fibers of type A, we know that given a Springer fiber, say $$\mathcal{B}_\lambda,$$ its irreducible components are all equidimensional and parametrized by standard Young tableaux of the Young diagram associated to partition $$\lambda$$.
Now the question: Given two standard Young Tableaux $$T_1^{\lambda}$$ and $$T_2^{\lambda},$$ of Young diagram $$\lambda,$$ is there a combinatorial way to find out whether the corresponding components, say $$\mathcal{B}^{T_1}_\lambda$$ and $$\mathcal{B}^{T_2}_\lambda,$$ have non-empty intersection?

• I think that one way to approach your question is using the torus fixed points in $\mathcal B_\lambda$.. The intersection of any two components is torus-invariant and thus if non-empty must contain a fixed point. We have a bijection between these torus fixed points and the set of row-strict (but not necc. column strict) tableaux of shape $\lambda$. If you can figure how for which row-strict tableaux $U$, we have $x_U \in \mathcal B_\lambda^{T}$, then you can answer the question. – Joel Kamnitzer Jul 2 '19 at 13:22
• Thank you @Joel. Now that I have finally understood your answer, I would ask the thing that you have mentioned: Is there a combinatorial algorithm that, given a standard tableau $T$, tells you for which row-strict tableau $U$ (except for ones whose standardization is T) $x_U \in \mathcal{B}_{\lambda}^T$ ? – Filip92 Apr 9 at 17:02