Universality of Rank 6 Plucker Relations and Grassmann Cone Preserving Maps

On Plücker Equations Characterizing Grassmann Cones

Polynomial solutions to the KP hierarchy are known to be parametrized by a cone over an infinite-dimensional Grassmann variety. Using the notion of Schubert derivation on a Grassmann algebra, we encode the classical Plücker equations of Grassmannians of r-dimensional subspaces in a formula whose limit for $r\rightarrow\infty$ coincides with the KP hierarchy phrased in terms of vertex operators.

On super Plücker embedding and cluster algebras

We define a super analog of the classical Plücker embedding of the Grassmannian into a projective space. One of the difficulties of the problem is rooted in the fact that super exterior powers $Λ^{r|s}(V)$ are not a simple generalization from the completely even case (this works only for $r|0$ when it is possible to use $Λ^r(V)$). To construct the embedding we need to non-trivially combine a super vector space $V$ and its parity-reversion $ΠV$. Our "super Plücker map" takes the Grassmann supermanifold $G_{r|s}(V)$ to a "weighted projective space" $P\left(Λ^{r|s}(V)\oplus Λ^{s|r}(ΠV)\right)$ wi…

A Grassmann algebra for matroids

We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plucker vector, which we view as a tensor, and a tropical linear space is recovered from its Plucker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plucker relations (valuated exchange axiom) if and only if the d-th wedge power of the kerne…

格拉斯曼流形的一般普吕克线性截面上的二次曲线模空间

Moduli of Conics on General Plucker Linear Sections of Grassmannians

该研究针对格拉斯曼流形的一般普吕克线性截面上的光滑二次曲线，通过相对舒伯特一般位置论证等方法，得到了不同余维线性截面上二次曲线希尔伯特概型的性质，并计算了特定情形下的二次曲线数量。

卷积滤波器子空间的格拉斯曼-普吕克参数化：正则性与闭嵌入

Grassmann--Plücker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings

该研究提出卷积滤波器子空间的格拉斯曼-普吕克参数化，证明其为闭嵌入，参数化有限双有理且纤维单元素，投影神经簇光滑，还通过Singular验证特例并探讨与滤波器冗余等的联系。

The equations defining affine Grassmannians in type A and a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman

The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called "shuffles", and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps: first we reinterpret the shuffles equations in terms of Frobenius twists of symmetr…