Richardson varieties are obtained as intersections of Schubert and opposite Schubert varieties. We provide a new family of toric degenerations of Richardson varieties inside Grassmannians by studying Gröbner degenerations of their corresponding ideals. These degenerations are parametrised by block diagonal matching fields in the sense of Sturmfels [33]. We associate a weight vector to each block diagonal matching field and study its corresponding initial ideal. In particular, we characterise when such ideals are toric, hence providing a family of toric degenerations for Richardson varieties. Given a Richardson variety \(X_{w}^v\) and a weight vector \(\mathbf{w}_\ell \) arising from a matching field, we consider two ideals: an ideal \(G_{k,n,\ell }|_w^v\) obtained by restricting the initial of the Plücker ideal to a smaller polynomial ring, and a toric ideal defined as the kernel of a monomial map \(\phi _\ell |_w^v\). We first characterise the monomial-free ideals of form \(G_{k,n,\ell }|_w^v\). Then we construct a family of tableaux in bijection with semi–standard Young tableaux which leads to a monomial basis for the corresponding quotient ring. Finally, we prove that when \(G_{k,n,\ell }|_w^v\) is monomial-free and the initial ideal in\(_{\mathbf{w}_\ell }(I(X_w^v))\) is quadratically generated, then all three ideals in\(_{\mathbf{w}_\ell }(I(X_w^v))\), \(G_{k,n,\ell }|_w^v\) and ker\((\phi _\ell |_w^v)\) coincide, and provide a toric degeneration of \(X_w^v\).

理查森（Richardson）变种是舒伯特与相反舒伯特变种的交集。通过研究相应理想理想的Gröbner变性，我们在格拉斯曼氏体内提供了Richardson变种的新的复曲面族。这些退化由Sturmfels的意义上的块对角线匹配场参数化[33]。我们将权重向量与每个块对角线匹配字段相关联，并研究其对应的初始理想值。特别是，我们表征了何时这些理想是复曲面的，从而为理查森品种提供了一个复曲面的退化家族。给定理查森变量\（X_ {w} ^ v \）和权重向量\（\ mathbf {w} _ \ ell \）来自匹配字段，我们考虑两个理想：理想\（G_ {k，n ，\ ell} | __w ^ v \）通过将Plücker理想的初始限制为较小的多项式环而获得，以及将复曲面理想限定为单项映射\（\ phi _ \ ell | _w ^ v \）的内核。我们首先描述形式\（G_ {k，n，\ ell} | _w ^ v \）的无单项式理想。然后，我们用半标准杨氏双射射流构造了一系列双射射流，从而为相应的商环带来了单项式。最终，我们证明当\（G_ {k，n，\ ell} | _w ^ v \）是无多项式的，并且在\（_ {\ mathbf {w} _ \ ell}（I（X_w ^ v））\）二次生成，然后\（_ {\ mathbf {w} _ \ ell}（I（X_w ^ v））\），\（G_ {k，n，\ ell} | _w ^ v \）和ker \（（\ phi _ \ ell | _w ^ v）\）一致，并提供\（X_w ^ v \）的复曲面退化。