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Multiplicity-free Schubert calculus. (English) Zbl 1210.14056
Let $$\text{Gr}(l, {\mathbb C}^n)$$ denote the Grassmannian of $$l$$-dimensional subspaces in $${\mathbb C}^n$$. The cohomology ring $$H^*(\text{Gr}(l, {\mathbb C}^n), \mathbb Z)$$ has an additive basis of Schubert classes $$\sigma_\lambda$$, indexed by Young diagrams $$\lambda = (\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_l \geq 0)$$ contained in the $$l \times k$$ rectangle where $$k = n - l$$ (which is denoted by $$\lambda \subset l\times k$$). The product of two Schubert classes in $$H^*({\text{Gr}}(l, \mathbb C^n), \mathbb Z)$$ is given by
$\sigma_\lambda \cdot \sigma_\mu =\sum_{ \nu \subset l\times k} c^\nu_{\lambda,\mu}\sigma_\nu ,$ where $$c^\nu_{\lambda,\mu}$$ is the classical Littlewood-Richardson coefficient. This expansion is multiplicity-free if $$c^\nu_{\lambda,\mu}\in \{0, 1\}$$ for all $$\nu\subset l\times k$$. In this paper, the authors give a nonrecursive, combinatorial answer to the following question of W. Fulton.
Question. When is $$\sigma_\lambda \cdot \sigma_\mu$$ multiplicity-free?
Their answer exploits the following considerations. For partitions $$\lambda, \mu \subset l\times k$$, place $$\lambda$$ against the upper left corner of the rectangle. Then rotate $$\mu$$ 180 degrees and place it in the lower right corner. The resulting subshape of $$l\times k$$ is referred to as $$\text{rotate}(\mu)$$.
If $$\lambda \cap \text{rotate}(\mu)\neq \emptyset$$, then the product $$\sigma_\lambda \cdot \sigma_\mu$$ is zero, and the interesting part is to handle the case of empty intersection. A Richardson quadruple is the datum $$(\lambda,\mu, l\times k)$$, where $$\lambda \cap \text{rotate}(\mu)= \emptyset$$. If $$\lambda \cup \text{rotate}(\mu)$$ does not contain a full $$l$$-column or $$k$$-row, call this Richardson quadruple basic. The main result of the article, Theorem 1.2, gives a criterion for multiplicity-freeness in terms of combinatorial conditions imposed on the basic Richardson quadruples.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 05E99 Algebraic combinatorics 14N15 Classical problems, Schubert calculus
##### Keywords:
Grassmannian; Richardson variety; Schubert class
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