This article presents a multimodal dataset comprising various representations and annotations of Franz Schubert’s song cycle Winterreise . seminal work constitutes an outstanding example the Romantic cycle—a central genre within Western classical music. Our unifies several public sources carefully created by music experts, compiled in comprehensive consistent way. The comprise singer’s lyrics, ...

We take the base field to be the field of complex numbers in these lectures. The varieties are, by definition, quasi-projective, reduced (but not necessarily irreducible) schemes. Let G be a semisimple, simply-connected, complex algebraic group with a fixed Borel subgroup B, a maximal torus H ⊂ B, and associated Weyl group W . (Recall that a Borel subgroup is any maximal connected, solvable sub...

For more than one hundred years, Schubert decomposition of complete and partial flag manifolds and Schubert calculus have been an intertwining point of algebra, geometry, representation theory, and combinatorics. In this paper, we are concerned with the further refinement of Schubert decomposition theory. Our far and ambitious goal is to describe the topology of intersection of two arbitrary Sc...

We prove that twisted versions of Schubert polynomials defined by $$\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$$ and S}_{ws_i} (s_i+\partial _i)\widetilde{\mathfrak S}_w$$ are monomial positive give a combinatorial formula for their coefficients. In doing so, we reprove extend previous result about positivity skew divided difference operators show how it implies the Pieri...

We consider linear codes associated to Schubert varieties in Grassmannians. A formula for the minimum distance of these codes was conjectured in 2000 and after having been established in various special cases, it was proved in 2008 by Xiang. We give an alternative proof of this formula. Further, we propose a characterization of the minimum weight codewords of Schubert codes by introducing the n...