In this paper, we describe a way to construct cycles which represent the Todd class of a toric va... more In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product, we assign a rational number μ ( σ ) \mu (\sigma ) to each rational polyhedral cone σ \sigma in the lattice, such that for any toric variety X X with fan Σ \Sigma in the lattice, we have \[ Td ( X ) = ∑ σ ∈ Σ μ ( σ ) [ V ( σ ) ] . \operatorname {Td}(X)=\sum _{\sigma \in \Sigma } \mu (\sigma ) [V(\sigma )]. \] This constitutes an improved answer to an old question of Danilov. In a similar way, beginning from the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli. Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by...
Using techniques introduced by H. Thomas and N. Williams in "Cyclic Symmetry of the Scaled S... more Using techniques introduced by H. Thomas and N. Williams in "Cyclic Symmetry of the Scaled Simplex," we prove that modular sweep maps are bijective. We construct the inverse of the modular sweep map by passing through an intermediary set of equitable partitions; motivated by an analogy to stable marriages, we prove that the set of equitable partitions for a fixed word forms a distributive lattice when ordered componentwise. We conclude that the general sweep maps defined by D. Armstrong, N. Loehr, and G. Warrington in "Sweep Maps: A Continuous Family of Sorting Algorithms" are bijective. As a special case of particular interest, this gives the first proof that the zeta map on rational Dyck paths is a bijection.
Let G be an acylic directed graph. For each vertex g ∈ G, we define an involution on the independ... more Let G be an acylic directed graph. For each vertex g ∈ G, we define an involution on the independent sets of G. We call these involutions flips, and use them to define a new partial order on independent sets of G. Trim lattices generalize distributive lattices by removing the graded hypothesis: a graded trim lattice is a distributive lattice, and every distributive lattice is trim. Our independence posets are a further generalization of distributive lattices, eliminating also the lattice requirement: an independence poset that is a lattice is always a trim lattice, and every trim lattice is the independence poset for a unique (up to isomorphism) acyclic directed graph G. We characterize when an independence poset is a lattice with a graph-theoretic condition on G. We generalize the definition of rowmotion from distributive lattices to independence posets, and we show it can be computed in three different ways. We also relate our constructions to torsion classes, semibricks, and 2-si...
A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector... more A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If $Q$ is a Dynkin quiver and $m$ is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including $m$ in their support, the category of which we denote by $\mathcal{C}_{Q,m}$, are determined up to isomorphism by this invariant. We use this invariant to define a bijection from isomorphism classes of representations in $\mathcal{C}_{Q,m}$ to reverse plane partitions whose shape is the minuscule poset corresponding to $Q$ and $m$. By relating the piecewise-linear promotion action on reverse plane partitions to Auslander-Reiten translation in the derived category, we give a uniform proof that the order of promotion equals the Co...
We use Khovanov and Kuperberg’s web growth rules to identify the leading term in the invariant as... more We use Khovanov and Kuperberg’s web growth rules to identify the leading term in the invariant associated to an $\textrm{SL}_3$ web diagram, with respect to a particular term order.
For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for ... more For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for $\operatorname{Hom}(X,\unicode[STIX]{x1D70F}Y)$. We use this to describe support $\unicode[STIX]{x1D70F}$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $\unicode[STIX]{x1D70F}$-tilting modules and functorially finite torsion classes. We give an explicit basis of $\operatorname{Ext}^{1}(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.
This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl gr... more This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton exte...
In this manuscript we show that two partial orders defined on the set of Littlewood-Richardson fi... more In this manuscript we show that two partial orders defined on the set of Littlewood-Richardson fillings of a~given shape $(\alpha,\beta,\gamma)$ are equivalent if $\beta\setminus\gamma$ is a horizontal and vertical strip. In fact, we give two proofs for the equivalence of the box order and the dominance order for fillings, both are algorithmic. The first of these proofs emphasizes links to the Bruhat order for the symmetric group and the second gives more straightforward construction of box-moves. This work is motivated by the known result that the equivalence of the two combinatorial orders leads to a description of the geometry of the representation space of invariant subspaces of nilpotent linear operators.
In this paper, we describe a way to construct cycles which represent the Todd class of a toric va... more In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product, we assign a rational number μ ( σ ) \mu (\sigma ) to each rational polyhedral cone σ \sigma in the lattice, such that for any toric variety X X with fan Σ \Sigma in the lattice, we have \[ Td ( X ) = ∑ σ ∈ Σ μ ( σ ) [ V ( σ ) ] . \operatorname {Td}(X)=\sum _{\sigma \in \Sigma } \mu (\sigma ) [V(\sigma )]. \] This constitutes an improved answer to an old question of Danilov. In a similar way, beginning from the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli. Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by...
Using techniques introduced by H. Thomas and N. Williams in "Cyclic Symmetry of the Scaled S... more Using techniques introduced by H. Thomas and N. Williams in "Cyclic Symmetry of the Scaled Simplex," we prove that modular sweep maps are bijective. We construct the inverse of the modular sweep map by passing through an intermediary set of equitable partitions; motivated by an analogy to stable marriages, we prove that the set of equitable partitions for a fixed word forms a distributive lattice when ordered componentwise. We conclude that the general sweep maps defined by D. Armstrong, N. Loehr, and G. Warrington in "Sweep Maps: A Continuous Family of Sorting Algorithms" are bijective. As a special case of particular interest, this gives the first proof that the zeta map on rational Dyck paths is a bijection.
Let G be an acylic directed graph. For each vertex g ∈ G, we define an involution on the independ... more Let G be an acylic directed graph. For each vertex g ∈ G, we define an involution on the independent sets of G. We call these involutions flips, and use them to define a new partial order on independent sets of G. Trim lattices generalize distributive lattices by removing the graded hypothesis: a graded trim lattice is a distributive lattice, and every distributive lattice is trim. Our independence posets are a further generalization of distributive lattices, eliminating also the lattice requirement: an independence poset that is a lattice is always a trim lattice, and every trim lattice is the independence poset for a unique (up to isomorphism) acyclic directed graph G. We characterize when an independence poset is a lattice with a graph-theoretic condition on G. We generalize the definition of rowmotion from distributive lattices to independence posets, and we show it can be computed in three different ways. We also relate our constructions to torsion classes, semibricks, and 2-si...
A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector... more A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If $Q$ is a Dynkin quiver and $m$ is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including $m$ in their support, the category of which we denote by $\mathcal{C}_{Q,m}$, are determined up to isomorphism by this invariant. We use this invariant to define a bijection from isomorphism classes of representations in $\mathcal{C}_{Q,m}$ to reverse plane partitions whose shape is the minuscule poset corresponding to $Q$ and $m$. By relating the piecewise-linear promotion action on reverse plane partitions to Auslander-Reiten translation in the derived category, we give a uniform proof that the order of promotion equals the Co...
We use Khovanov and Kuperberg’s web growth rules to identify the leading term in the invariant as... more We use Khovanov and Kuperberg’s web growth rules to identify the leading term in the invariant associated to an $\textrm{SL}_3$ web diagram, with respect to a particular term order.
For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for ... more For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for $\operatorname{Hom}(X,\unicode[STIX]{x1D70F}Y)$. We use this to describe support $\unicode[STIX]{x1D70F}$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $\unicode[STIX]{x1D70F}$-tilting modules and functorially finite torsion classes. We give an explicit basis of $\operatorname{Ext}^{1}(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.
This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl gr... more This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton exte...
In this manuscript we show that two partial orders defined on the set of Littlewood-Richardson fi... more In this manuscript we show that two partial orders defined on the set of Littlewood-Richardson fillings of a~given shape $(\alpha,\beta,\gamma)$ are equivalent if $\beta\setminus\gamma$ is a horizontal and vertical strip. In fact, we give two proofs for the equivalence of the box order and the dominance order for fillings, both are algorithmic. The first of these proofs emphasizes links to the Bruhat order for the symmetric group and the second gives more straightforward construction of box-moves. This work is motivated by the known result that the equivalence of the two combinatorial orders leads to a description of the geometry of the representation space of invariant subspaces of nilpotent linear operators.
For A a gentle algebra, and X and Y string modules, we construct a combinatorial basis for Hom(X,... more For A a gentle algebra, and X and Y string modules, we construct a combinatorial basis for Hom(X, τ Y). We use this to describe support τ-tilting modules for A. We give a combinatorial realization of maps in both directions realizing the bijection between support τ-tilting modules and func-torially finite torsion classes. We give an explicit basis of Ext 1 (Y, X) as short exact sequences.
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Papers by Hugh Thomas