# Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties

@article{Bonala2020StandardMT, title={Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties}, author={Narasimha Chary Bonala and Oliver Clarke and Fatemeh Mohammadi}, journal={arXiv: Algebraic Geometry}, year={2020} }

We study toric degenerations of opposite Schubert and Richardson varieties inside degenerations of Grassmannians and flag varieties. These degenerations are parametrized by matching fields in the sense of Sturmfels and Zelevinsky. We construct so-called restricted matching field ideals whose generating sets are understood combinatorially through tableaux. We determine when these ideals are toric and coincide with Grobner degenerations of Richardson varieties using the well established standard… Expand

#### 3 Citations

Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux

- Mathematics, Computer Science
- J. Symb. Comput.
- 2021

An analogue of matching field ideals for Schubert varieties inside the flag variety and a complete characterization of toric ideals among them are described to show that block diagonal matching fields give rise to toric degenerations. Expand

Toric degenerations of flag varieties from matching field tableaux

- Mathematics
- 2019

We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Plucker algebras.… Expand

Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux

- Mathematics
- 2020

Abstract We study Grobner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in… Expand

#### References

SHOWING 1-10 OF 38 REFERENCES

Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux

- Mathematics, Computer Science
- J. Symb. Comput.
- 2021

An analogue of matching field ideals for Schubert varieties inside the flag variety and a complete characterization of toric ideals among them are described to show that block diagonal matching fields give rise to toric degenerations. Expand

Standard monomial theory for desingularized Richardson varieties in the flag variety GL(n)/B

- Mathematics
- 2011

We consider a desingularization Γ of a Richardson variety in the variety of complete flags, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. For any projective embedding of… Expand

Toric degenerations of flag varieties from matching field tableaux

- Mathematics
- 2019

We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Plucker algebras.… Expand

Following Schubert varieties under Feigin's degeneration of the flag variety

- Mathematics
- 2018

We describe the effect of Feigin's flat degeneration of the type $\textrm{A}$ flag variety on its Schubert varieties. In particular, we study when they stay irreducible and in several cases we are… Expand

Degenerations of Flag and Schubert Varieties to Toric Varieties

- 1996

In this paper, we prove the degenerations of Schubert varieties in a minuscule G=P , as well as the class of Kempf varieties in the ag variety SL(n)=B, to (normal) toric varieties. As a consequence,… Expand

Toric degenerations of Grassmannians from matching fields

- Mathematics
- 2018

We study the algebraic combinatorics of monomial degenerations of Plucker forms which is governed by matching fields in the sense of Sturmfels and Zelevinsky. We provide a necessary condition for a… Expand

On toric degenerations of flag varieties

- Mathematics
- 2017

Following the historical track in pursuing TT-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation… Expand

Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

- Mathematics
- 2005

In the sl_n case, A. Berenstein and A. Zelevinsky studied the Schutzenberger involution in terms of Lusztig's canonical basis, [3]. We generalize their construction and formulas for any semisimple… Expand

Toric degenerations of Schubert varieties

- Mathematics
- 2000

LetG be a simply connected semisimple complex algebraic group. We prove that every Schubert variety ofG has a flat degeneration into a toric variety. This provides a generalization of results of [9],… Expand

Toric degeneration of Schubert varieties and Gelfand¿Tsetlin polytopes

- Mathematics
- 2003

This note constructs the flat toric degeneration of the manifold Fln of flags in Cn due to Gonciulea and Lakshmibai (Transform. Groups 1(3) (1996) 215) as an explicit GIT quotient of the Grobner… Expand