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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.

14M15 Grassmannians, Schubert varieties, flag manifolds
05E99 Algebraic combinatorics
14N15 Classical problems, Schubert calculus
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