arXiv subjects

Konstanze Rietsch

Publications and source records attributed to Konstanze Rietsch.

At least 19 recordsLinked to original sources

arXiv2026

Grassmannian quantum cohomology in the infinite limit and total positivity

The theory of total positivity was shown by Lusztig to be intrinsically linked to the canonical basis with its positivity properties. When we restrict ourselves to studying total positivity just for the set of lower-triangular unipotent Toeplitz matrices, say in type $A$, then there is a similar link with the quantum cohomology rings of flag varieties and the Schubert bases and their positivity properties. Namely, this builds on a theory of Dale Peterson that gives a uniform Lie-theoretic description of all of the quantum cohomology rings $qH^*(G/P)$. In a precursor to this paper, the Schubert basis and quantum parameters in $qH^*(SL_n/B)$, which restrict to positive-valued functions on totally positive Toeplitz matrices, were analysed with respect to their limiting behaviour as $n\to\infty$, uncovering a novel connection with the classical Edrei theorem on parametrising the infinite totally positive Toeplitz matrices. In this paper we study the Grassmannian case, using the conventions from the $SL_{n}/B$ setting as a guide, and we determine the quantum parameter and Schubert class asymptotics in different scenarios. Along the way, we obtain a new interpretation of the strange duality involution on the localised quantum cohomolgy ring of the Grassmannian. Finally, we prove an asymptotic formula for quantum parameters in a partial flag setting, and we furthermore formulate some conjectures concerning partial flag varieties and related quantum cohomology asymptotics.

math.CO↗

arXiv2025

Totally positive Toeplitz matrices: classical and modern

By a theorem of Edrei, an infinite, normalised totally nonnegative upper-triangular Toeplitz matrix is determined by a pair of nonnegative parameter sequences, the `Schoenberg parameters', where nonzero parameters correspond to the roots and poles of a naturally associated generating function. These totally nonnegative Toeplitz matrices and their parameters also arise in the classification of characters of the infinite symmetric group by later work of Thoma. Moreover the Schoenberg parameters have an asymptotic interpretation in terms of irreducible representations of S_n and their Young diagrams by Vershik-Kerov. In this article we consider infinite totally positive Toeplitz matrices as limits of finite ones, and we obtain two further asymptotic descriptions of the Schoenberg parameters that are now related to quantum cohomology of the flag variety as n goes to infinity. One is related to asymptotics of normalised quantum parameters, and the other to asymptotics of the Chern classes of the tautological line bundles. We also describe the asymptotics of (quantum) Schubert classes in terms of the Schoenberg parameters. Our limit formulas relate to and were motivated by a tropical analogue of this theory that we survey. In the tropical setting one finds an asymptotic relationship between the `tropical Schoenberg parameters' and the weight map from Lusztig's parametrisation of the canonical basis.

math.CO↗

arXiv2025

Tropical Toeplitz matrices and parametrisations

The set of infinite upper-triangular totally positive Toeplitz matrices has a classical parametrisation proved by Edrei et al and originally conjectured by Schoenberg, that involves pairs of sequences of positive real parameters. These matrices (and their parameters) are central for understanding characters of the infinite symmetric group by work of Thoma. On the other hand there is a very different parametrisation theorem that applies to the finite analogue of this set. These finite Toeplitz matrices and their parameters relate to quantum cohomology of flag varieties and mirror symmetry. In this paper we replace the positive reals by a semifield with valuation to then construct tropical analogues for both parametrisation theorems. In the finite case we tropicalise using positive generalised Puiseaux series. This builds on work of Judd and Lüdenbach. In the infinite case we use a new valued semifield of continuous functions. We arrive at different natural infinite analogues of totally positive Toeplitz matrices, depending on a choice of topology on our valued semifield. We then prove an asymptotic result relating the tropical parameters from the finite case to the tropicalisations of the Schoenberg parameters. Moreover, we show that our finite type tropical parametrisation map is given by Lusztig's weight map from the theory of canonical bases. This results in a surprising connection between the classical Edrei theorem with its Schoenberg parameters and Lusztig's canonical basis parametrisation.

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arXiv2025

The tropical critical point and mirror symmetry

Call a Laurent polynomial $W$ `complete' if its Newton polytope is full-dimensional with zero in its interior. We show that if $W$ is any complete Laurent polynomial with coefficients in the positive part of the field $K$ of generalised Puiseux series, then $W$ has a unique positive critical point $p_{crit}$. Here a generalised Puiseux series is called `positive' if the coefficient of its leading term is in $\mathbb R_{>0}$. Using the valuation on $K$ we obtain a canonically associated `tropical critical point' $d_{crit}$ in $\mathbb R^{r}$ for which we give a finite recursive construction. We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also give applications to toric geometry including, via the theory of [FOOO], to the construction of canonical non-displaceable Lagrangian tori for toric symplectic manifolds.

math.AG↗

arXiv2025

A superpotential for Grassmannian Schubert varieties

While mirror symmetry for flag varieties and Grassmannians has been extensively studied, Schubert varieties in the Grassmannian are singular, and hence standard mirror symmetry statements are not well-defined. Nevertheless, in this article we introduce a ``superpotential'' $W^λ$ for each Grassmannian Schubert variety $X_λ$, generalizing the Marsh-Rietsch superpotential for Grassmannians, and we show that $W^λ$ governs many toric degenerations of $X_λ$. We also generalize the ``polytopal mirror theorem'' for Grassmannians from our previous work: namely, for any cluster seed $G$ for $X_λ$, we construct a corresponding Newton-Okounkov convex body $Δ_G^λ$, and show that it coincides with the superpotential polytope $Γ_G^λ$, that is, it is cut out by the inequalities obtained by tropicalizing an associated Laurent expansion of $W^λ$. This gives us a toric degeneration of the Schubert variety $X_λ$ to the (singular) toric variety $Y(\mathcal{N}_λ)$ of the Newton-Okounkov body. Finally, for a particular cluster seed $G=G^λ_{\mathrm{rec}}$ we show that the toric variety $Y(\mathcal{N}_λ)$ has a small toric desingularisation, and we describe an intermediate partial desingularisation $Y(\mathcal{F}_λ)$ that is Gorenstein Fano. Many of our results extend to more general varieties in the Grassmannian.

math.AG↗

arXiv2025

An anticanonical perspective on G/P Schubert varieties

We describe a natural basis of the Cartier class group of an arbitrary Schubert variety $X_{w,P}$ in a flag variety $G/P$ of general Lie type. We then characterise when the Schubert variety is factorial/Fano, along with an explicit formula for the anticanonical line bundle in these cases. We also prove that, for Schubert varieties in simply-laced types (only), being factorial is equivalent to being $Q$-factorial, and is equivalent to the equality of the Betti numbers $b_2(X_{w,P})=b_{2\ell(w)-2}(X_{w,P})$. Finally, we give a convenient characterisation of when a simply-laced Schubert variety is Gorenstein and when it is Gorenstein Fano.

math.AG↗

arXiv2025

Root polytopes, flow polytopes, and order polytopes

In this paper we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb{R}^n$, where $e_1,\dots,e_n$ is the standard basis of $\mathbb{R}^n$. Such a polytope can be encoded by a quiver $Q$ with vertices $V \subseteq \{v_1,\dots,v_n\} \cup \{\star\}$, where each edge $v_j\to v_i$ or $\star \to v_i$ or $v_i\to \star$ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\operatorname{Root}(Q)$. These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver $Q$ is strongly-connected then the root polytope $\operatorname{Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\operatorname{Root}(Q)$. We also show that if $Q$ is planar, then $\operatorname{Root}(Q)$ is (integrally equivalent to the) polar dual of the flow polytope of the dual quiver. Finally we consider the case that $Q$ comes from a ranked poset $P$, and show that $\operatorname{Root}(Q)$ is polar dual to (a translation of) a marked poset polytope. We then study the toric variety $Y(\mathcal{F}_Q)$ associated to the face fan $\mathcal{F}_Q$ of $\operatorname{Root}(Q)$. If $Q$ comes from a ranked poset $P$ we give a combinatorial description of the Picard group of $Y(\mathcal{F}_Q)$, and we show that $Y(\mathcal{F}_Q)$ is a small partial desingularisation of the Hibi toric variety $Y_{\mathcal{O}(P)}$ of the order polytope $\mathcal{O}(P)$. We show that $Y(\mathcal{F}_Q)$ has a small crepant toric resolution of singularities $Y(\widehat{\mathcal{F}}_Q)$, and as a consequence that the Hibi toric variety $Y_{\mathcal{O}(P)}$ has a small resolution of singularities for any ranked poset $P$. These results have applications to mirror symmetry.

math.CO↗

arXiv2024

A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus

We construct a Plücker coordinate superpotential $\mathcal{F}_-$ that is mirror to a partial flag variety $\mathbb{ F}\ell(n_\bullet)$. Its Jacobi ring recovers the small quantum cohomology of $\mathbb{ F}\ell(n_\bullet)$ and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues for the action of the first Chern class $c_1(\mathbb{ F}\ell(n_\bullet))$ on quantum cohomology are equal to the critical values of $\mathcal{F}_-$. We achieve this by proving new identities in quantum Schubert calculus that are inspired by our formula for $\mathcal{F}_-$ and the mirror symmetry conjecture.

math.AG↗

arXiv2024

Generalisations of Euler's Tonnetz on triangulated surfaces

We give a definition of a what we call a `tonnetz' on a triangulated surface, generalising the famous tonnetz of Euler from 1739. In Euler's tonnetz the vertices of a regular `$A_2$ triangulation' of the plane are labelled with notes, or pitch-classes. In our generalisation we allow much more general labellings of triangulated surfaces. In particular, edge labellings turn out to lead to a rich set of examples. We construct natural examples that are related to crystallographic reflection groups and live on triangulations of tori. Underlying these we observe a curious relationship between mathematical Langlands duality and major/minor duality. We also construct `exotic' type-$A_2$ examples (different from Euler's Tonnetz), and a tonnetz on a sphere that encodes all major ninth chords.

math.CO↗

arXiv2021

Planar spider theorem and asymmetric Frobenius algebras

The `spider theorem' for a general Frobenius algebra $A$, classifies all maps $A^{\otimes m}\to A^{\otimes n}$ that are built from the operations and, in a graphical representation, represented by a {\it connected} diagram. Here the algebra can be noncommutative and the Frobenius form can be asymmetric. We view this theorem as reducing any connected diagram to a standard form with $j$ beads $B$, where $j$ is the number of bounded connected components of the original diagram. We study the associated F-dimension Hilbert series $\dim_x=\sum_{j=0}^\infty x^j\dim_j$, where $\dim_j=ε\circ B^j\circ 1$ are invariants of the Frobenius structure. We also study moduli of asymmetric quasispecial and `weakly symmetric' Frobenius structures and their F-dimensions. Examples include general Frobenius structures on matrix algebras $A=M_d(k)$ and on group algebras $k G$ as well as on $u_q(sl_2)$ at low roots of unity.

math.QA↗

arXiv2020

The B-model connection and mirror symmetry for Grassmannians

We consider the Grassmannian X of (n-k)-dimensional subspaces of an n-dimensional complex vector space. We describe a `mirror dual' Landau-Ginzburg model for X consisting of the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian together with a superpotential expressed succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to the one proposed by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990s by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to the superpotential a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of X. We also prove a T-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology D-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's J-function, which was conjectured in a 1998 paper by Batyrev, Ciocan-Fontaine, Kim and van Straten.

math.AG↗

arXiv2019

Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian $X=Gr_{n-k}(\mathbb C^n)$, as well as the mirror dual Landau-Ginzburg model $(\check{X}^\circ, W_q:\check{X}^\circ \to \mathbb C)$, where $\check{X}^\circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{X} = Gr_k((\mathbb C^n)^*)$, and the superpotential W_q has a simple expression in terms of Plücker coordinates. Grassmannians simultaneously have the structure of an $\mathcal{A}$-cluster variety and an $\mathcal{X}$-cluster variety. Given a cluster seed G, we consider two associated coordinate systems: a $\mathcal X$-cluster chart $Φ_G:(\mathbb C^*)^{k(n-k)}\to X^{\circ}$ and a $\mathcal A$-cluster chart $Φ_G^{\vee}:(\mathbb C^*)^{k(n-k)}\to \check{X}^\circ$. To each $\mathcal X$-cluster chart $Φ_G$ and ample `boundary divisor' $D$ in $X\setminus X^{\circ}$, we associate a Newton-Okounkov body $Δ_G(D)$ in $\mathbb R^{k(n-k)}$, which is defined as the convex hull of rational points. On the other hand using the $\mathcal A$-cluster chart $Φ_G^{\vee}$ on the mirror side, we obtain a set of rational polytopes, described by inequalities, by writing the superpotential $W_q$ in the $\mathcal A$-cluster coordinates, and then "tropicalising". Our main result is that the Newton-Okounkov bodies $Δ_G(D)$ and the polytopes obtained by tropicalisation coincide. As an application, we construct degenerations of the Grassmannian to toric varieties corresponding to these Newton-Okounkov bodies. Additionally, when $G$ corresponds to a plabic graph, we give a formula for the lattice points of the Newton-Okounkov bodies, which has an interpretation in terms of quantum Schubert calculus.

math.AG↗

arXiv2017

Cluster duality and mirror symmetry for Grassmannians

In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. For our $A$-model, we consider the Grassmannian $\mathbb X=Gr_{n-k}(\mathbb{C}^n)$. The $B$-model is a Landau-Ginzburg model $(\check{\mathbb X}^\circ, W_q:\check{\mathbb X}^\circ \to \mathbb{C})$, where $\check{\mathbb X}^\circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{\mathbb X} = Gr_k((\mathbb{C}^n)^*)$, and the superpotential $W_q$ has a simple expression in terms of Plücker coordinates, see [MarshRietsch]. From a given plabic graph $G$ we obtain two coordinate systems: using work of Postnikov and Talaska we have a positive chart $Φ_G:(\mathbb{C}^*)^{k(n-k)}\to \mathbb X$ in our $A$-model, and using work of Scott we have a cluster chart $Φ_G^{\vee}:(\mathbb{C}^*)^{k(n-k)}\to \check{\mathbb X}$ in our $B$-model. To each positive chart $Φ_G$ and choice of positive integer $r$, we associate a polytope $NO_G^r$, which we construct as the convex hull of a set of integer lattice points. This polytope is an example of a Newton-Okounkov polytope associated to the line bundle $\mathcal O(r)$ on $\mathbb X$. On the other hand, using the cluster chart $Φ_G^{\vee}$ and the same positive integer $r$, we obtain a polytope $Q_G^r$ -- described in terms of inequalities -- by "tropicalizing" the composition $W_{t^r}\circ Φ_G^{\vee}$. Our main result is that the polytopes $NO_G^r$ and $Q_G^r$ coincide.

math.AG↗

arXiv2014

On Landau-Ginzburg models for quadrics and flat sections of Dubrovin connections

This paper proves a version of mirror symmetry expressing the (small) Dubrovin connection for even-dimensional quadrics in terms of a mirror-dual Landau-Ginzburg model (Xcan,W). Here Xcan is the complement of an anticanonical divisor in a Langlands dual quadric. The superpotential W is a regular function on Xcan and is written in terms of coordinates which are naturally identified with a cohomology basis of the original quadric. This superpotential is shown to extend the earlier Landau-Ginzburg model of Givental, and to be isomorphic to the Lie-theoretic mirror introduced by Rietsch. We also introduce a Laurent polynomial superpotential which is the restriction of W to a particular torus in Xcan. Together with results of Pech-Rietsch for odd quadrics, we obtain a combinatorial model for the Laurent polynomial superpotential in terms of a quiver, in the vein of those introduced in the 1990's by Givental for type A full flag varieties. These Laurent polynomial superpotentials form a single series, despite the fact that our mirrors of even quadrics are defined on dual quadrics, while the mirror to an odd quadric is naturally defined on a projective space. Finally, we express flat sections of the (dual) Dubrovin connection in a natural way in terms of oscillating integrals associated to (Xcan,W) and compute explicitly a particular flat section.

math.AG↗

arXiv2012

Lie theory of finite simple groups and the Roth property

In noncommutative geometry a `Lie algebra' or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset C stable under inversion. We study the associated Killing form. For the universal calculus associated to C=G \ {e} we show that the magnitude of the Killing form μ=\sum_{a,b\in C}K^{-1}_{a,b} is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ is not equal to 1/(N-1), where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N-1)-dimensional subspace of invariant vectors is invertible iff the finite group is almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known that most finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the generating conjugacy class in the case of the dihedral groups D_{2n} with n odd has invertible Killing form, and the same for the 2-cycles conjugacy class in any S_n. We also compute some eigenvalues of the Killing form in the case of the n-cycles class in S_n. Finally, we verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

math.QA↗

arXiv2012

Lie theory and coverings of finite groups

We introduce the notion of an `inverse property' (IP) quandle C which we propose as the right notion of `Lie algebra' in the category of sets. To any IP quandle we construct an associated group G_C. For a class of IP quandles which we call `locally skew' and when G_C is finite we show that the noncommutative de Rham cohomology H^1(G_C) is trivial aside from a single generator θthat has no classical analogue. If we start with a group G then any subset C\subseteq G\setminus {e} which is ad-stable and inversion-stable naturally has the structure of an IP quandle. If C also generates G then we show that G_C \twoheadrightarrow G with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that G_C\twoheadrightarrow G is an isomorphism for all finite crystallographic reflection groups W with C the set of reflections, and that C is locally skew precisely in the simply laced case. This implies that H^1(W)=k when W is simply laced, proving in particular a previous conjecture for S_n. We obtain similar results for the dihedral groups D_{6m}. We also consider C=Z P^1\cup Z P^1 as a locally skew IP-quandle `Lie algebra' of SL_2(Z) and show that G_C\cong B_3, the braid group on 3 strands. The map B_3\twoheadrightarrow SL_2(Z) which arises naturally as a covering map in our theory, coincides with the restriction of the universal covering map \widetilde {SL_2(R)}\to SL_2(R) to the inverse image of SL_2(Z).

math.QA↗

arXiv2012

Total positivity, Schubert positivity, and Geometric Satake

Let G be a simple and simply-connected complex algebraic group, and let X \subset G^\vee be the centralizer subgroup of a principal nilpotent element. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian Gr_G. Peterson furthermore connected this ring to the quantum cohomology rings of partial flag varieties G/P. The first aim of this paper is to study three different notions of positivity on X: (1) Schubert positivity arising via Peterson's work, (2) total positivity in the sense of Lusztig, and (3) Mirkovic-Vilonen positivity obtained from the MV-cycles in Gr_G. Our first main theorem establishes that these three notions of positivity coincide. The second aim of this paper is to parametrize the totally nonnegative part of X, confirming a conjecture of the second author. In type A a substantial part of our results were previously established by the second author. The crucial new component of this paper is the connection with the affine Grassmannian and the geometric Satake correspondence.

math.RT↗

arXiv2011

A mirror symmetric solution to the quantum Toda lattice

We use representation theory to construct integral formulas for solutions to the quantum Toda lattice in general type. This result generalizes work of Givental for SL(n)/B in a uniform way to arbitrary type and can be interpreted as a kind of mirror theorem for the full flag variety G/B. We also prove the existence of a totally positive critical point of the 'superpotential' in every mirror fiber.

math.RT↗