<   Oliver Pechenik: Algebraic Combinatorics and Schubert Calculus   >

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1. Degrees of P-Grothendieck polynomials and regularity of Pfaffian varieties (with Matthew St.Denis)
   
   Submitted, 24 pages.
   arXiv:2405.17645
   

   
   We prove a formula for the degrees of Ikeda and Naruse's P-Grothendieck polynomials using combinatorics of shifted tableaux. We show this formula can be used in conjunction with results of Hamaker, Marberg, and Pawlowski to obtain an upper bound on the Castelnuovo–Mumford regularity of certain Pfaffian varieties known as vexillary skew-symmetric matrix Schubert varieties. imilar combinatorics additionally yields a new formula for the degree of Grassmannian Grothendieck polynomials and the regularity of Grassmannian matrix Schubert varieties, complementing a 2021 formula of Rajchgot, Ren, Robichaux, St. Dizier, and Weigandt.

2. Web bases in degree two from hourglass plabic graphs (with Christian Gaetz, Stephan Pfannerer, Jessica Striker and Joshua Swanson)
   
   Submitted, 24 pages.
   arXiv:2402.13978
   

   
   Webs give a diagrammatic calculus for spaces of \(U_q(\mathfrak{sl}_r)\)-tensor invariants, but intrinsic characterizations of web bases are only known in certain cases. Recently, we introduced hourglass plabic graphs to give the first such \(U_q(\mathfrak{sl}_4)\)-web bases. Separately, Fraser introduced a web basis for Plücker degree two representations of arbitrary \(U_q(\mathfrak{sl}_r)\). Here, we show that Fraser's basis agrees with that predicted by the hourglass plabic graph framework and give an intrinsic characterization of the resulting webs. A further compelling feature with many applications is that our bases exhibit rotation-invariance. Together with the results of our earlier paper, this implies that hourglass plabic graphs give a uniform description of all known rotation-invariant \(U_q(\mathfrak{sl}_r)\)-web bases. Moreover, this provides a single combinatorial model simultaneously generalizing the Tamari lattice, the alternating sign matrix lattice, and the lattice of plane partitions. As a part of our argument, we develop properties of square faces in arbitrary hourglass plabic graphs, a key step in our program towards general \(U_q(\mathfrak{sl}_r)\)-web bases.

3. Rotation-invariant web bases from hourglass plabic graphs (with Christian Gaetz, Stephan Pfannerer, Jessica Striker and Joshua Swanson)
   
   Submitted, 65 pages.
   arXiv:2306.12501
   

   
   Webs give a diagrammatic calculus for spaces of tensor invariants. We introduce hourglass plabic graphs as a new avatar of webs, and use these to give the first rotation-invariant \(U_q(\mathfrak{sl}_4)\)-web basis, a long-sought object. The characterization of our basis webs relies on the combinatorics of these new plabic graphs and associated configurations of a symmetrized six-vertex model. We give growth rules, based on a novel crystal-theoretic technique, for generating our basis webs from tableaux and we use skein relations to give an algorithm for expressing arbitrary webs in the basis. We also discuss how previously known rotation-invariant web bases can be unified in our framework of hourglass plabic graphs.

4. The Kromatic symmetric function: A \(K\)-theoretic analogue of \(X_G\) (with Logan Crew and Sophie Spirkl)
   
   Submitted, 15 pages.
   arXiv:2301.02177
   📹 video (part 1) and video (part 2) by Logan Crew
   

   
   Schur functions are a basis of the symmetric function ring that represent Schubert cohomology classes for Grassmannians. Replacing the cohomology ring with \(K\)-theory yields a rich combinatorial theory of inhomogeneous deformations, where Schur functions are replaced by their \(K\)-analogues, the basis of symmetric Grothendieck functions. We introduce and initiate a theory of the Kromatic symmetric function \(\overline{X}_G\), a \(K\)-theoretic analogue of the chromatic symmetric function \(X_G\) of a graph \(G\). The Kromatic symmetric function is a generating series for graph colorings in which vertices may receive any nonempty set of distinct colors such that neighboring color sets are disjoint.
   
   Our main result lifts a theorem of Gasharov (1996) to this setting, showing that when \(G\) is a claw-free incomparability graph, \(\overline{X}_G\) is a positive sum of symmetric Grothendieck functions. This result suggests a topological interpretation of Gasharov's theorem. We then show that the Kromatic symmetric functions of path graphs are not positive in any of several \(K\)-analogues of the e-basis of symmetric functions, demonstrating that the Stanley-Stembridge conjecture (1993) does not have such a lift to \(K\)-theory and so is unlikely to be amenable to a topological perspective. We also define a vertex-weighted extension of \(\overline{X}_G\) and show that it admits a deletion–contraction relation. Finally, we give a \(K\)-analogue for \(\overline{X}_G\) of the classic monomial-basis expansion of \(X_G\).

5. Quasisymmetric Schubert calculus (with Matthew Satriano)
   
   Submitted, 32 pages.
   arXiv:2205.12415
   📹 video
   

   
   The ring of symmetric functions occupies a central place in algebraic combinatorics, with a particularly notable role in Schubert calculus, where the standard cell decompositions of Grassmannians yield the celebrated family of Schur functions and the cohomology ring is governed by Littlewood-Richardson rules. The past 50 years have seen an analogous development of quasisymmetric function theory, with applications to enumerative combinatorics, Hopf algebras, graph theory, representation theory, and other areas. Despite such successes, this theory has lacked a quasisymmetric analogue of Schubert calculus. In particular, there has been much interest, since work of Lam and Pylyavskyy (2007), in developing "K-theoretic" analogues of quasisymmetric function theory, for which a major obstacle has been the lack of topological interpretations. Here, building on work of Baker and Richter (2008), we apply the philosophy of Schubert calculus to the loop space \(\Omega(\Sigma(\mathbb{CP}^\infty))\) through the homotopy model given by James reduced product \(J(\mathbb{CP}^\infty)\). We describe a canonical Schubert cell decomposition of \(J(\mathbb{CP}^\infty)\), yielding a canonical basis of its cohomology, which we explicitly identify with monomial quasisymmetric functions. Our constructions apply equally to James reduced products of generalized flag varieties \(G/P\), and we show how Littlewood-Richardson rules for any \(G/P\) lift to \(H^\star(J(G/P))\). If \(J(\mathbb{CP}^\infty)\) carried the structure of a normal projective algebraic variety, the structure sheaves of the cell closures would yield a "cellular K-theory" Schubert basis. We show this is impossible. Nonetheless, we introduce and study a more subtle K-theory Schubert basis. We characterize this K-theory ring and develop quasisymmetric representatives with an explicit combinatorial description.

6. Web invariants for flamingo Specht modules (with Chris Fraser, Rebecca Patrias and Jessica Striker)
   
   Algebraic Combinatorics, to appear, 2024, 31 pages.
   arXiv:2308.07256
   

   
   Webs yield an especially important realization of certain Specht modules, irreducible representations of symmetric groups, as they provide a pictorial basis with a convenient diagrammatic calculus. In recent work, the last three authors associated polynomials to noncrossing partitions without singleton blocks, so that the corresponding polynomials form a web basis of the pennant Specht module \(S^{(d,d,1n−2d)}\). These polynomials were interpreted as global sections of a line bundle on a 2-step partial flag variety. Here, we both simplify and extend this construction. On the one hand, we show that these polynomials can alternatively be situated in the homogeneous coordinate ring of a Grassmannian, instead of a 2-step partial flag variety, and can be realized as tensor invariants of classical (but highly nonplanar) tensor diagrams. On the other hand, we extend these ideas from the pennant Specht module \(S^{(d,d,1n−2d)}\) to more general flamingo Specht modules \(S^{(dr,1n−rd)}\). In the hook case \(r=1\), we obtain a spanning set that can be restricted to a basis in various ways. In the case \(r>2\), we obtain a basis of a well-behaved subspace of \(S^{(dr,1n−rd)}\), but not of the entire module.

7. An inverse Grassmannian Littlewood-Richardson rule and extensions (with Anna Weigandt)
   
   Forum of Mathematics, Sigma, to appear, 2024, 24 pages.
   arXiv:2202.11185
   📹 video by Anna Weigandt
   

   
   Schubert structure coefficients \(c^w_{u,v}\) describe the multiplicative structure of the cohomology rings of flag varieties. Much work has been done on the problem of giving combinatorial formulas for these coefficients in special cases. In particular, the Littlewood-Richardson rule computes \(c^w_{u,v}\) in the case that \(u\), \(v\), and \(w\) are all \(p\)-Grassmannian permutations for some common \(p\). Building on work on Wyser (2013), we introduce backstable clans to prove a "dual" positive combinatorial rule that computes \(c^w_{u,v}\) when \(u^{−1}\) is \(p\)-Grassmannian and \(v^{-1}\) is \(q\)-Grassmannian. We derive new families of linear relations among Schubert structure coefficients, which we then use to give a further positive combinatorial rule for \(c^w_{u,v}\) in the case that \(u^{-1}\) is \(p\)-Grassmannian and \(v^{-1}\) is covered in weak Bruhat order by a \(q\)-Grassmannian permutation.

8. Promotion permutations for tableaux (with Christian Gaetz, Stephan Pfannerer, Jessica Striker and Joshua Swanson)
   
   Combinatorial Theory 4(2), Article No. 15, 2024, 56 pages.
   arXiv:2306.12506
   

   
   In our companion paper, we develop a new \(\mathrm{SL}_4\)-web basis. Basis elements are given by certain planar graphs and are constructed so that important algebraic operations can be performed diagrammatically. A guiding principle behind our construction is that the long cycle \((12\ldots n) \in \mathfrak{S}_n\) should act by rotation of webs. Moreover, the bijection between webs and tableaux should intertwine rotation with the promotion action on tableaux.
   
   In this paper, we develop necessary notions of promotion permutations and promotion matrices, which are new even for standard tableaux. To support inductive arguments in the companion paper, we must however work in the more general setting of fluctuating tableaux, which we introduce and which subsumes many classes of tableaux that have been previously studied, including (generalized) oscillating, vacillating, rational, alternating, and (semi)standard tableaux. Therefore, we also give here a full development of the basic combinatorics and representation theory of fluctuating tableaux.

9. Proof of a conjectured Möbius inversion formula for Grothendieck polynomials (with Matthew Satriano)
   
   Selecta Mathematica 30, Article No. 83, 2024, 8 pages.
   arXiv:2202.02897
   

   
   Schubert polynomials \(\mathfrak{S}_w\) are polynomial representatives for cohomology classes of Schubert varieties in a complete flag variety, while Grothendieck polynomials \(\mathfrak{G}_w\) are analogous representatives for the \(K\)-theory classes of the structure sheaves of Schubert varieties. In the special case that \(\mathfrak{S}_w\) is a multiplicity-free sum of monomials, K. Mészáros, L. Setiabrata, and A. St. Dizier conjectured that \(\mathfrak{G}_w\) can be easily computed from \(\mathfrak{S}_w\) via Möbius inversion on a certain poset. We prove this conjecture. Our approach is to realize monomials as Chow classes on a product of projective spaces and invoke a result of M. Brion on flat degenerations of such classes.

10. Castelnuovo–Mumford regularity of matrix Schubert varieties (with David Speyer and Anna Weigandt)
    
    Selecta Mathematica 30, Article No. 66, 2024, 44 pages.
    arXiv:2111.10681
    📹 video
    

    
    Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo–Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo–Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo–Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo–Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order.

11. James reduced product schemes and double quasisymmetric functions (with Matthew Satriano)
    
    Advances in Mathematics 449, Paper No. 109737, 2024, 28 pages.
    arXiv:2304.11508
    📹 video
    

    
    Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that \(\mathrm{QSym}\) manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product \(J\mathbb{C}\mathbb{P}^\infty\). In recent work, we used this viewpoint to develop topologically-motivated bases of \(\mathrm{QSym}\) and initiate a Schubert calculus for \(J\mathbb{C}\mathbb{P}^\infty\) in both cohomology and \(K\)-theory.
    
    Here, we study the torus-equivariant cohomology of \(J\mathbb{C}\mathbb{P}^\infty\). We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood–Richardson rule for the structure coefficients of this basis.
    
    Our main geometric tool is the GKM theory of Goresky, Kottwitz, and MacPherson (1998). To apply this tool, we first introduce an algebro-geometric analogue of the James reduced product construction, showing that James reduced products can be realized as projective varieties. Our projective variety has too many invariant curves for GKM theory to apply directly; we resolve this by establishing a general result on equivariant cohomology of varieties with affine pavings, which allows us to exchange tori.

12. Tableau evacuation and webs (with Rebecca Patrias)
    
    Proceedings of the American Mathematical Society. Series B 10, 2023, 341-352.
    arXiv:2109.04989
    

    
    Webs are certain planar diagrams embedded in disks. They index and describe bases of tensor products of representations of \(\mathfrak{sl}_2\) and \(\mathfrak{sl}_3\). There are explicit bijections between webs and certain rectangular tableaux. Work of Petersen-Pylyavskyy-Rhoades (2009) and Russell (2013) shows that these bijections relate web rotation to tableau promotion. We describe the analogous relation between web reflection and tableau evacuation.

13. A web basis of invariant polynomials from noncrossing partitions (with Rebecca Patrias and Jessica Striker)
    
    Advances in Mathematics 408, Paper No. 108603, 2022, 33 pages.
    arXiv:2112.05781
    📹 video (part 1) and video (part 2)
    

    
    The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules \(S^\lambda\). Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases \(\lambda=(d,d)\) and \(\lambda=(d,d,d)\). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with \(\lambda\) of the form \((d,d,1^\ell)\). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.

14. K-theoretic crystals for set-valued tableaux of rectangular shapes (with Travis Scrimshaw)
    
    Algebraic Combinatorics 5(3), 2022, 515-536.
    arXiv:1904.09674
    

    
    In earlier work with C. Monical (2021), we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard set-valued tableaux of any fixed rectangular shape. Here, we establish this conjecture by explicitly constructing the K-crystal operators. As a consequence, we establish the first combinatorial formula for Lascoux polynomials \(L_{w \lambda}\) when \(\lambda\) is a multiple of a fundamental weight as the sum over flagged set-valued tableaux. Using this result, we then prove corresponding cases of conjectures of Ross-Yong (2015) and Monical (2016) by constructing bijections with the respective combinatorial objects.

15. Curious cyclic sieving on increasing tableaux (with Christian Gaetz, Jessica Striker and Joshua Swanson)
    
    Enumerative Combinatorics and Applications 2(3), Article #S2R18, 2022, 8 pages.
    arXiv:2112.09228
    

    
    We prove a cyclic sieving result for the set of \(3 \times k\) packed increasing tableaux with maximum entry \(m:=3+k\) under K-promotion. The "curiosity" is that the sieving polynomial arises from the \(q\)-hook formula for standard tableaux of "toothbrush shape" \((2^3,1^{k−2})\) with \(m+1\) boxes, whereas K-promotion here only has order \(m\).

16. Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass
    
    Combinatorial Theory 2(1), Paper No. 15, 2022, 20 pages.
    arXiv:2107.02679
    

    
    Let \(P\) be a graded poset of rank \(r\) and let \(\mathbf{c}\) be a \(c\)-element chain. For an order ideal \(I\) of \(P \times \mathbf{c}\), its rowmotion \(\psi(I)\) is the smallest ideal containing the minimal elements of the complementary filter of \(I\). The map \(\psi\) defines invertible dynamics on the set of ideals. We say that that P has NRP ("not relatively prime") rowmotion if no \(\psi\)-orbit has cardinality relatively prime to \(r+c+1\).
    
    In work with R. Patrias (2020), we proved a 1995 conjecture of P. Cameron and D. Fon-Der-Flaass by establishing NRP rowmotion for the product \(P= \mathbf{a} \times \mathbf{b}\) of two chains, the poset whose order ideals correspond to the Schubert varieties of a Grassmann variety \(\mathrm{Gr}_a(\mathbb{C}^{a+b})\) under containment. Here, we initiate the general study of posets with NRP rowmotion.
    
    Our first main result establishes NRP rowmotion for all minuscule posets \(P\), posets whose order ideals reflect the Schubert stratification of minuscule flag varieties. Our second main result is that NRP promotion depends only on the isomorphism class of the comparability graph of \(P\).

17. Gröbner geometry of Schubert polynomials through ice (with Zachary Hamaker and Anna Weigandt)
    
    Advances in Mathematics 398, Paper No. 108228, 2022, 29 pages.
    arXiv:2003.13719
    📹 video
    

    
    The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of "vexillary" matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of Lascoux (2002) in terms of the 6-vertex ice model (recently rediscovered by Lam, Lee, and Shimozono (2018) in the guise of "bumpless pipe dreams").
    
    Our Conjectures 7.1 and 7.2 have been proven by Patricia Klein. The most important part of our main Conjecture 2.7 has been proven by Klein and Anna Weigandt, while the full statement remains open.

18. Crystal structures for symmetric Grothendieck polynomials (with Cara Monical and Travis Scrimshaw)
    
    Transformation Groups 26(3), 2021, 1025-1075.
    arXiv:1807.03294
    

    
    The symmetric Grothendieck polynomials representing Schubert classes in the K-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type \(A_n\) crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand–Tsetlin patterns via the 5-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials.

19. Path-cordial abelian groups (with Rebecca Patrias)
    
    Australasian Journal of Combinatorics 80(1), 2021, 157-166.
    arXiv:2006.13764
    

    
    A labeling of the vertices of a graph by elements of any abelian group \(A\) induces a labeling of the edges by summing the labels of their endpoints. Hovey defined the graph \(G\) to be \(A\)-cordial if it has such a labeling where the vertex labels and the edge labels are both evenly-distributed over \(A\) in a technical sense. His conjecture that all trees \(T\) are \(A\)-cordial for all cyclic groups \(A\) remains wide open, despite significant attention. Curiously, there has been very little study of whether Hovey's conjecture might extend beyond the class of cyclic groups.
    
    We initiate this study by analyzing the larger class of finite abelian groups \(A\) such that all path graphs are \(A\)-cordial. We conjecture a complete characterization of such groups, and establish this conjecture for various infinite families of groups as well as for all groups of small order.
    
    Our conjectures have since been proven by Sylwia Cichacz.

20. Polynomials from combinatorial \(K\)-theory (with Cara Monical and Dominic Searles)
    
    Canadian Journal of Mathematics 73(1), 2021, 29-62.
    arXiv:1806.03802
    

    
    We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasiLascoux basis, which is simultaneously both a \(K\)-theoretic deformation of the quasikey basis and also a lift of the \(K\)-analogue of the quasiSchur basis from quasisymmetric polynomials to general polynomials. We give positive expansions of this quasiLascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasiLascoux basis. As a special case, these expansions give the first proof that the \(K\)-analogues of quasiSchur polynomials expand positively in multifundamental quasisymmetric polynomials of T. Lam and P. Pylyavskyy.
    
    The second new basis is the kaon basis, a \(K\)-theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis.
    
    Throughout, we explore how the relationships among these \(K\)-analogues mirror the relationships among their cohomological counterparts. We make several 'alternating sum' conjectures that are suggestive of Euler characteristic calculations.

21. Dynamics of plane partitions: Proof of the Cameron–Fon-Der-Flaass conjecture (with Rebecca Patrias)
    
    Forum of Mathematics, Sigma 8, 2020, article e62, 6 pages.
    arXiv:2003.13152
    📹 video
    

    
    One of the oldest outstanding problems in dynamical algebraic combinatorics is the following conjecture of P. Cameron and D. Fon-Der-Flaass (1995). Consider a plane partition \(P\) in an \(a × b × c\) box \({\mathbf B}\). Let \(\Psi(P)\) denote the smallest plane partition containing the minimal elements of \(\mathbf{B} - P\). Then if \(p = a+b+c − 1\) is prime, Cameron and Fon-Der-Flaass conjectured that the cardinality of the \(\Psi\)-orbit of \(P\) is always a multiple of \(p\). This conjecture was established for \(p \gg 0\) by Cameron and Fon-Der-Flaass (1995) and for slightly smaller values of \(p\) in work of K. Dilks, J. Striker, and the second author (2017). Our main theorem specializes to prove this conjecture in full generality.

22. Asymmetric function theory (with Dominic Searles)
    
    Schubert Calculus and Its Applications in Combinatorics and Representation Theory, Springer, 2020, 73-112.
    Proceedings of the International Festival in Schubert Calculus (Guangzhou, China, 2017).
    arXiv:1904.01358
    

    
    The classical theory of symmetric functions has a central position in algebraic combinatorics, bridging aspects of representation theory, combinatorics, and enumerative geometry. More recently, this theory has been fruitfully extended to the larger ring of quasisymmetric functions, with corresponding applications. Here, we survey recent work extending this theory further to general asymmetric polynomials.

23. Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley (with Zachary Hamaker, David Speyer, and Anna Weigandt)
    
    Algebraic Combinatorics 3(2), 2020, 301-307.
    arXiv:1812.00321
    📹 video
    

    
    We study the action of a differential operator on Schubert polynomials. Using this action, we first give a short new proof of an identity of I. Macdonald (1991). We then prove a determinant conjecture of R. Stanley (2017). This conjecture implies the (strong) Sperner property for the weak order on the symmetric group, a property recently established by C. Gaetz and Y. Gao (2019).

24. The genomic Schur function is fundamental-positive
    
    Annals of Combinatorics 24(1), 2020, 95-108.
    arXiv:1810.04727
    

    
    In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood-Richardson coefficients in torus-equivariant K-theory of Grassmannians. We then studied the genomic Schur function \(U_\lambda\), a generating function for such tableaux, showing that it is non-trivially a symmetric function, although generally not Schur-positive. Here we show that \(U_\lambda\) is, however, positive in the basis of fundamental quasisymmetric functions. We give a positive combinatorial formula for this expansion in terms of gapless increasing tableaux; this is, moreover, the first finite expression for \(U_\lambda\). Combined with work of A. Garsia and J. Remmel, this yields a compact combinatorial (but necessarily non-positive) formula for the Schur expansion of \(U_\lambda\).

25. Doppelgängers: Bijections of plane partitions (with Zachary Hamaker, Rebecca Patrias, and Nathan Williams)
    
    International Mathematics Research Notices 2020(2), 2020, 487-540.
    arXiv:1602.05535
    Slides
    

    
    We say two posets are "doppelgängers" if they have the same number of \(P\)-partitions of each height \(k\). We give a uniform framework for bijective proofs that posets are doppelgängers by synthesizing \(K\)-theoretic Schubert calculus techniques of H. Thomas and A. Yong with M. Haiman's rectification bijection and an observation of R. Proctor. Geometrically, these bijections reflect the rational equivalence of certain subvarieties of minuscule flag manifolds. As a special case, we provide the first bijective proof of a 1983 theorem of R. Proctor—that plane partitions of height \(k\) in a rectangle are equinumerous with plane partitions of height \(k\) in a trapezoid.

26. Decompositions of Grothendieck polynomials (with Dominic Searles)
    
    International Mathematics Research Notices 2019(10), 2019, 3214-3241.
    arXiv:1611.02545
    Slides
    

    
    Finding a combinatorial rule for the Schubert structure constants in the \(K\)-theory of flag varieties is a long-standing problem. The Grothendieck polynomials of Lascoux and Schützenberger (1982) serve as polynomial representatives for \(K\)-theoretic Schubert classes, but no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in \(n\) variables), give a positive combinatorial formula for the expansion of Grothendieck polynomials in these "glide polynomials," and provide a positive combinatorial Littlewood-Richardson rule for expanding a product of Grothendieck polynomials in the glide basis.
    
    A specialization of the glide basis recovers the fundamental slide polynomials of Assaf and Searles (2016), which play an analogous role with respect to ordinary cohomology. Additionally, the stable limits of another specialization of glide polynomials are Lam and Pylyavskyy's (2007) basis of multi-fundamental quasisymmetric functions, \(K\)-theoretic analogues of Gessel's (1984) fundamental quasisymmetric functions. Those glide polynomials that are themselves quasisymmetric are truncations of multi-fundamental quasisymmetric functions and form a basis of quasisymmetric polynomials.

27. Unique rectification in \(d\)-complete posets: Towards the \(K\)-theory of Kac-Moody flag varieties (with Rahul Ilango^† and Michael Zlatin^†)
    
    Electronic Journal of Combinatorics 25(4), 2018, 1-35.
    arXiv:1805.02287
    

    
    This paper is the result of an REU project from Summer 2017 with two Rutgers University undergraduates.
    
    The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of "unique rectification targets" in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to \(K\)-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor’s "\(d\)-complete posets" to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac- Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of \(d\)-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain \(K\)-theoretic Schubert structure constants in the Kac-Moody setting.

28. Orbits of plane partitions of exceptional Lie type (with Holly Mandel)
    
    European Journal of Combinatorics 74, 2018, 90-109.
    arXiv:1712.09180
    Slides
    

    
    For each minuscule flag variety \(X\), there is a corresponding minuscule poset, describing its Schubert decomposition. We study an action on plane partitions over such posets, introduced by P. Cameron and D. Fon-der-Flaass (1995). For plane partitions of height at most 2, D. Rush and X. Shi (2013) proved an instance of the cyclic sieving phenomenon, completely describing the orbit structure of this action. They noted their result does not extend to greater heights in general; however, when \(X\) is one of the two minuscule flag varieties of exceptional Lie type \(E\), they conjectured explicit instances of cyclic sieving for all heights.
    
    We prove their conjecture in the case that \(X\) is the Cayley-Moufang plane of type \(E_6\). For the other exceptional minuscule flag variety, the Freudenthal variety of type \(E_7\), we establish their conjecture for heights at most 4, but show that it fails generally. We further give a new proof of an unpublished cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height in the case \(X\) is an even-dimensional quadric hypersurface. Our argument uses ideas of K. Dilks, O. Pechenik, and J. Striker (2017) to relate the action on plane partitions to combinatorics derived from \(K\)-theoretic Schubert calculus.

29. Deformed cohomology of flag varieties (with Dominic Searles)
    
    Mathematical Research Letters, 25, 2018, 649-657.
    arXiv:1410.8070
    

    
    This paper introduces a two-parameter deformation of the cohomology of generalized flag varieties. One special case is the Belkale-Kumar deformation (used to study eigencones of Lie groups). Another picks out intersections of Schubert varieties that behave nicely under projections. Our construction yields a new proof that the Belkale-Kumar product is well-defined. This proof is shorter and more elementary than earlier proofs.

30. Rhombic tilings and Bott-Samelson varieties (with Laura Escobar, Bridget Tenner, and Alexander Yong)
    
    Proceedings of the American Mathematical Society 146, 2018, 1921-1935.
    arXiv:1605.05613
    Slides
    

    
    S. Elnitsky (1997) gave an elegant bijection between rhombic tilings of \(2n\)-gons and commutation classes of reduced words in the symmetric group on \(n\) letters. P. Magyar (1998) found an important construction of the Bott-Samelson varieties introduced by H.C. Hansen (1973) and M. Demazure (1974). We explain a natural connection between S. Elnitsky's and P. Magyar's results. This suggests using tilings to encapsulate Bott-Samelson data (in type \(A\)). It also indicates a geometric perspective on S. Elnitsky's combinatorics. We also extend this construction by assigning desingularizations to the zonotopal tilings considered by B. Tenner (2006).

31. Promotion of increasing tableaux: Frames and homomesies
    
    Electronic Journal of Combinatorics 24(3), 2017, 1-14.
    arXiv:1702.01358
    

    
    A key fact about M.-P. Schützenberger's (1972) promotion operator on rectangular standard Young tableaux is that iterating promotion once per entry recovers the original tableau. For tableaux with strictly increasing rows and columns, H. Thomas and A. Yong (2009) introduced a theory of \(K\)-jeu de taquin with applications to \(K\)-theoretic Schubert calculus. The author (2014) studied a \(K\)-promotion operator \(\mathcal{P}\) derived from this theory, but observed that this key fact does not generally extend to \(K\)-promotion of such increasing tableaux.
    
    Here, we show that the key fact holds for labels on the boundary of the rectangle. That is, for \(T\) a rectangular increasing tableau with entries bounded by \(q\), we have \(\mathrm{Frame}(\mathcal{P}^{q}(T)) = \mathrm{Frame}(T)\), where \(\mathrm{Frame}(U)\) denotes the restriction of \(U\) to its first and last row and column. Using this fact, we obtain a family of homomesy results on the average value of certain statistics over \(K\)-promotion orbits, extending a 2-row theorem of J. Bloom, D. Saracino, and the author (2016) to arbitrary rectangular shapes.

32. Genomic tableaux (with Alexander Yong)
    
    Journal of Algebraic Combinatorics 45, 2017, 649-685.
    arXiv:1603.08490
    

    
    We explain how genomic tableaux [Pechenik-Yong `15] are a semistandard complement to increasing tableaux [Thomas-Yong `09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender-Knuth involutions, as well as Schur functions from (shifted) semistandard Young tableaux theory. These are applied to obtain new Littlewood-Richardson rules for \(K\)-theory Schubert calculus of Grassmannians (after [Buch `02]) and maximal orthogonal Grassmannians (after [Clifford- Thomas-Yong `14], [Buch-Ravikumar `12]). For the unsolved case of Lagrangian Grassmannians, sharp upper and lower bounds using genomic tableaux are conjectured.

33. Resonance in orbits of plane partitions and increasing tableaux (with Kevin Dilks and Jessica Striker)
    
    Journal of Combinatorial Theory, Series A 148, 2017, 244-274.
    arXiv:1512.00365
    Slides
    

    
    We introduce a new concept of resonance on discrete dynamical systems. This concept formalizes the observation that, in various combinatorially-natural cyclic group actions, orbit cardinalities are all multiples of divisors of a fundamental frequency. Our prototypical example of this phenomenon is B. Wieland's gyration action on alternating sign matrices. Our main result is an equivariant bijection between plane partitions in a box (or order ideals in the product of three chains) under rowmotion and increasing tableaux under \(K\)-promotion. Both of these actions were observed to have orbit sizes that were small multiples of divisors of an expected orbit size, and we show this is an instance of resonance, as \(K\)- promotion cyclically rotates the set of labels appearing in the increasing tableaux. We extract a number of corollaries from this equivariant bijection, including a strengthening of a theorem of [P. Cameron–D. Fon-der-Flaass '95] and several new results on the order of \(K\)-promotion. Along the way, we adapt the proof of the conjugacy of promotion and rowmotion from [J. Striker–N. Williams '12] to give a generalization in the setting of n-dimensional lattice projections.

34. Equivariant K-theory of Grassmannians II: The Knutson-Vakil conjecture (with Alexander Yong)
    
    Compositio Mathematica 153, 2017, 667-677.
    arXiv:1508.00446
    

    
    In 2005, A. Knutson–R. Vakil conjectured a puzzle rule for equivariant \(K\)-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors’ recently proved tableau rule for the same Schubert calculus problem.

35. Equivariant K-theory of Grassmannians (with Alexander Yong)
    
    Forum of Mathematics, Pi 5, 2017, article e3, 128 pages.
    arXiv:1506.01992
    

    
    We address a unification of the Schubert calculus problems solved by [A. Buch ’02] and [A. Knutson-T. Tao ’03]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant \(K\)-theory of Grassmannians with respect to the basis of Schubert structure sheaves. We thereby deduce the conjectural rule of [H. Thomas- A. Yong ’13] for the same coefficients. Both rules are positive in the sense of [D. Anderson- S. Griffeth-E. Miller ’11] (and moreover in a stronger form). Our work is based on the combinatorics of genomic tableaux and a generalization of [M.-P. Sch¨utzenberger ’77]’s jeu de taquin.

36. K-theoretic Schubert calculus and applications
    
    Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2016, 255 pages.
    

    
    A central result in algebraic combinatorics is the Littlewood-Richardson rule that governs products in the cohomology of Grassmannians. A major theme of the modern Schubert calculus is to extend this rule and its associated combinatorics to richer cohomology theories. This thesis focuses on \(K\)-theoretic Schubert calculus. We prove the first Littlewood-Richardson rule in torus-equivariant \(K\)-theory. We thereby deduce the conjectural rule of H. Thomas and A. Yong, as well as a mild correction to the conjectural rule of A. Knutson and R. Vakil. Our rule manifests the positivity established geometrically by D. Anderson, S. Griffeth and E. Miller, and moreover in a stronger `squarefree' form that resolves an issue raised by A. Knutson. Our work is based on the combinatorics of genomic tableaux, which we introduce, and a generalization of M.-P. Schützenberger's jeu de taquin. We further apply genomic tableaux to obtain new rules in non-equivariant \(K\)-theory for Grassmannians and maximal orthogonal Grassmannians, as well as to make various conjectures relating to Lagrangian Grassmannians. This is joint work with Alexander Yong. Our theory of genomic tableaux is a semistandard analogue of the increasing tableau theory initiated by H. Thomas and A. Yong. These increasing tableaux carry a natural lift of M.-P. Schützenberger's promotion operator. We study the orbit structure of this action, generalizing a result of D. White by establishing an instance of the cyclic sieving phenomenon of V. Reiner, D. Stanton and D. White. In joint work with J. Bloom and D. Saracino, we prove a homomesy conjecture of J. Propp and T. Roby for promotion on standard tableaux, which partially generalizes to increasing tableaux. In joint work with K. Dilks and J. Striker, we relate the action of \(K\)-promotion on increasing tableaux to the rowmotion operator on plane partitions, yielding progress on a conjecture of P. Cameron and D. Fon-der-Flaass. Building on this relation between increasing tableaux and plane partitions, we apply the \(K\)-theoretic jeu de taquin of H. Thomas and A. Yong to give, in joint work with Z. Hamaker, R. Patrias and N. Williams, the first bijective proof of a 1983 theorem of R. Proctor, namely that that plane partitions of height \(k\) in a rectangle are equinumerous with plane partitions of height \(k\) in a trapezoid.

37. Proofs and generalizations of a homomesy conjecture of Propp and Roby (with Jonathan Bloom and Dan Saracino)
    
    Discrete Mathematics 339, 2016, 194-206.
    arXiv:1308.0546
    

    
    Let \(G\) be a group acting on a set \(X\) of combinatorial objects, with finite orbits, and consider a statistic \(\xi \colon X \rightarrow \mathbb{C} \). Propp and Roby defined the triple \((X, G, \xi )\) to be homomesic if for any orbits \(\mathcal{O}_1, \mathcal{O}_2\), the average value of the statistic \(\xi\) is the same, that is \[ \frac{1}{|\mathcal{O}_1|} \sum_{x \in \mathcal{O}_1} \xi(x) = \frac{1}{|\mathcal{O}_2|} \sum_{y \in \mathcal{O}_2} \xi(y).\] In 2013 Propp and Roby conjectured the following instance of homomesy. Let \(\mathrm{SSYT}_k(m \times n)\) denote the set of semistandard Young tableaux of shape \(m \times n\) with entries bounded by \(k\). Let \(S\) be any set of boxes in the \(m \times n\) rectangle fixed under 180° rotation. For \(T \in \mathrm{SSYT}_k(m \times n)\), define \(\sigma_S(T)\) to be the sum of the entries of \(T\) in the boxes of \(S\). Let \(\langle P \rangle\) be a cyclic group of order \(k\) where \(P\) acts on \(\mathrm{SSYT}_k(m \times n)\) by promotion. Then \((\mathrm{SSYT}_k(m \times n), \langle P \rangle, \sigma_S )\) is homomesic.
    We prove this conjecture, as well as a generalization to cominuscule posets. We also discuss analogous questions for tableaux with strictly increasing rows and columns under the \(K\)-promotion of Thomas and Yong, and prove limited results in that direction.

38. Genomic tableaux and combinatorial K-theory (with Alexander Yong)
    
    Discrete Mathematics and Theoretical Computer Science Proceedings FPSAC`15, 2015, 37-48. FPSAC 2015, Daejeon, Korea.
    

    
    We introduce genomic tableaux, with applications to Schubert calculus. We report a combinatorial rule for structure coefficients in the torus-equivariant \(K\)-theory of Grassmannians for the basis of Schubert structure sheaves. This rule is positive in the sense of [Anderson-Griffeth-Miller ’11]. We thereby deduce an earlier conjecture of [Thomas-Yong ’13] for the coefficients. Moreover, our rule specializes to give a new Schubert calculus rule in the (non-equivariant) \(K\)-theory of Grassmannians. From this perspective, we also obtain a new rule for \(K\)-theoretic Schubert structure constants of maximal orthogonal Grassmannians, and give conjectural bounds on such constants for Lagrangian Grassmannians.
    
    This paper is an extended abstract of items 12 and 16 above.

39. Cyclic sieving of increasing tableaux and small Schröder paths
    
    Journal of Combinatorial Theory, Series A 125, 2014, 357-378.
    arXiv:1209.1355
    

    
    An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. We demonstrate relations between the jeu de taquin for increasing tableaux developed by H. Thomas and A. Yong and the combinatorics of tropical frieze patterns. We then use this jeu de taquin to present new instances of the cyclic sieving phenomenon of V. Reiner, D. Stanton, and D. White, generalizing results of D. White and of J. Stembridge.

40. Generalized graph cordiality (with Jennifer Wise)
    
    Discussiones Mathematicae Graph Theory 32(3), 2012, 557-567.
    

    
    Hovey introduced \(A\)-cordial labelings as a simultaneous generalization of cordial and harmonious labelings. If \(A\) is an abelian group, then a labeling \(f \colon V(G) \rightarrow A\) of the vertices of some graph \(G\) induces an edge-labeling on \(G\); the edge \(uv\) receives the label \(f(u) + f(v)\). A graph \(G\) is \(A\)-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one. Research on \(A\)-cordiality has focused on the case where \(A\) is cyclic. In this paper, we investigate \(V_4\)-cordiality of many families of graphs, namely complete bipartite graphs, paths, cycles, ladders, prisms, and hypercubes. We find that all complete bipartite graphs are \(V_4\)-cordial except \(K_{m,n}\) where \(m, n \equiv 2 \pmod 4\). All paths are \(V_4\)-cordial except \(P_4\) and \(P_5\). All cycles are \(V_4\)-cordial except \(C_4\), \(C_5\), and \(C_k\), where \(k \equiv 2 \pmod 4\). All ladders \(P_2 \square P_k\) are \(V_4\)-cordial except \(C_4\). All prisms are \(V_4\)-cordial except \(P_2 \square C_k\), where \(k \equiv 2 \pmod 4\). All hypercubes are \(V_4\)-cordial, except \(C_4\). Finally, we introduce a generalization of \(A\)-cordiality involving digraphs and quasigroups, and we show that there are infinitely many \(Q\)-cordial digraphs for every quasigroup \(Q\).

41. A midsummer knot's dream (with Allison Henrich, Noel MacNaughton, Sneha Narayan, Robert Silversmith, and Jennifer Townsend)
    
    College Mathematics Journal 42(2), 2011, 126-134.
    arXiv:1003.4494
    

    
    In this paper, we introduce playing games on shadows of knots. We demonstrate two novel games, namely, To Knot or Not to Knot and Much Ado about Knotting. We also discuss winning strategies for these games on certain families of knot shadows. Finally, we suggest variations of these games for further study.

42. Classical and virtual pseudodiagram theory and new bounds on unknotting numbers and genus (with Allison Henrich, Noel MacNaughton, Sneha Narayan, and Jennifer Townsend)
    
    Journal of Knot Theory and Its Ramifications 20(4), 2011, 625-650.
    arXiv:0908.1981
    

    
    A pseudodiagram is a diagram of a knot with some crossing information missing. We review and expand the theory of pseudodiagrams introduced by R. Hanaki. We then extend this theory to the realm of virtual knots, a generalization of knots. In particular, we investigate how much crossing information must be known to conclude that a diagram is a diagram of the unknot (the trivializing number). We also consider how much information is necessary to identify a non-trivial knot, a classical knot, or a non-classical knot. We then apply pseudodiagram theory to develop new upper bounds on unknotting number, virtual unknotting number, and genus.

43. Large cardinals
    
    Undergraduate Honors Thesis, Oberlin College, 2010
    

    
    Infinite sets are a fundamental object of modern mathematics. Surprisingly, the existence of infinite sets cannot be proven within mathematics. Their existence, or even the consistency of their possible existence, must be justified extra-mathematically or taken as an article of faith. We describe here several varieties of large infinite set that have a similar status in mathematics to that of infinite sets, i.e. their existence cannot be proven, but they seem both reasonable and useful. These large sets are known as large cardinals. We focus on two types of large cardinal: inaccessible cardinals and measurable cardinals. Assuming the existence of a measurable cardinal allows us to disprove a questionable statement known as the Axiom of Constructibility (\(V=L\)).

44. Zero-divisor ideals and realizable zero-divisor graphs (with Michael Axtell, Joe Stickles, et al.)
    
    Involve 2(1), 2009, 17-27.
    

    
    We seek to classify the sets of zero-divisors that form ideals based on their zero-divisor graphs. We offer full classification of these ideals within finite commutative rings with identity. We also provide various results concerning the realizability of a graph as a zero-divisor graph.
    
    This paper came out of three summers of an REU at Wabash College. I participated in the third of these, and so never met a number of the other authors!

45. The nilradical and non-nilradical graphs of commutative rings (with Abigail Bishop, Thomas Cuchta and Kathryn Boddie)
    
    International Journal of Algebra 2(20), 2008, 981-994.
    

    
    We introduce two subgraphs of the zero-divisor graph of a commutative ring \(R\): the nilradical and non-nilradical graphs. We examine their connective properties, diameter, and girth in relation to the algebraic properties of \(R\).
    
    This paper came out of an REU at Wabash College during Summer 2008.

46. Effects of temperature, salinity, and air exposure on development of the estuarine pulmonate gastropod Amphibola crenata (with Islay Marsden and Jan Pechenik)
    
    Journal of Experimental Marine Biology and Ecology 292(2), 2003, 159-176.
    

    
    The primitive pulmonate snail Amphibola crenata embeds embryos within a smooth mud collar on exposed estuarine mudflats in New Zealand. Development through hatching of free-swimming veliger larvae was monitored at 15 salinity and temperature combinations covering the range of 2– 30 ppt salinity and 15–25 °C. The effect of exposure to air on developmental rate was also assessed. There were approximately 18,000 embryos in each egg collar. The total number of veligers released from standard-sized egg collar fragments varied with both temperature and salinity: embryonic survival was generally higher at 15 and 20 °C than at 25 °C; moreover, survival was generally highest at intermediate salinities, and greatly reduced at 2 ppt salinity regardless of temperature. Even at 2 ppt salinity, however, about one-third of embryos were able to develop successfully to hatching. Embryonic tolerance to low salinity was apparently a property of the embryos themselves, or of the surrounding egg capsules; there was no indication that the egg collars protected embryos from exposure to environmental stress. Mean hatching times ranged between 7 and 22 days, with reduced developmental rates both at lower temperature and lower salinity. At each salinity tested, developmental rate to hatching was similar at 20 and 25 °C. At 15 °C, time to hatching was approximately double that recorded at the two higher exposure temperatures. Exposing the egg collars to air for 6–9 h each day at 20 °C (20 ppt salinity) accelerated hatching by about 24 h, suggesting that developmental rate in this species is limited by the rates at which oxygen or wastes can diffuse into and from intact collars, respectively. Similarly, veligers from egg capsules that were artificially separated from egg collars at 20 °C developed faster than those within intact egg collars. The remarkable ability of embryos of A. crenata to hatch over such a wide range of temperatures and salinities, and to tolerate a considerable degree of exposure to air, explains the successful colonization of this species far up into New Zealand estuaries.

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