Publications and preprints

With Nantel Bergeron, arXiv:coming.soon. Slides from Online Schubert Seminar

Submitted 2021. arXiv:2109.05651

With Anne Dranowski and Nicolle Gonzalez, Algebras and Representation Theory, to appear. arXiv:2210.10236

Abstract: Demazure crystals are subcrystals of highest weight irreducible g-crystals. In this article, we study tensor products of a larger class of subcrystals, called extremal, and give a local characterization for exactly when the tensor product of Demazure crystals is extremal. We then show that tensor products of Demazure crystals decompose into direct sums of Demazure crystals if and only if the tensor product is extremal, thus providing a sufficient and necessary local criterion for when the tensor product of Demazure crystals is itself Demazure. As an application, we show that the primary component in the tensor square of any Demazure crystal is always Demazure.

With Danjoseph Quijada, Electronic Journal of Combinatorics, to appear. arXiv:1908.08502

Abstract: Key polynomials are characters of Demazure modules for the general linear group that generalize the Schur polynomials. We prove a nonsymmetric generalization of Monk’s rule by giving a cancellation-free, multiplicity-free formula for the key polynomial expansion of the product of an arbitrary key polynomial with a degree one key polynomial.

With Sam Armon, Grant Bowling, Henry Ehrhard, J. Algebra 617 (2023) pp. 352–381. arXiv:2012.05382

Abstract: Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and gives a representation theoretic interpretation for Kohnert polynomials.

Combinatorial Theory 2.1 (2022), 9pp. arXiv:2003.01211

Abstract: Kohnert proposed a formula for Schubert polynomials as the generating polynomial for certain unit cell diagrams obtained from the diagram of a permutation. Billey, Jockusch and Stanley proved a formula for Schubert polynomials as the generating polynomial for compatible sequences of reduced words. In this paper, we give an explicit bijection between these two models, thereby proving Kohnert’s rule for Schubert polynomials.

Transactions of the AMS 375.3 (2022), pp. 2147–2186. arXiv:2002.07107

Abstract: Kohnert polynomials are polynomials indexed by unit cell diagrams in the first quadrant defined earlier by the author and Searles that give a common generalization of Schubert polynomials and Demazure characters for the general linear group. Demazure crystals are certain truncations of normal crystals whose characters are Demazure characters. For each diagram satisfying a southwest condition, we construct a Demazure crystal whose character is the Kohnert polynomial for the given diagram, resolving an earlier conjecture of the author and Searles that these polynomials expand nonnegatively into Demazure characters. We give explicit formulas for the expansions with applications including a characterization of those diagrams for which the corresponding Kohnert polynomial is a single Demazure character.

Journal of Combinatorics 14.1 (2023), pp. 21–51. arXiv:2103.05207

Abstract: We introduce a new paradigm for proving the Schur P-positivity of a given quasi-symmetric function. Generalizing dual equivalence, we give an axiomatic definition for a family of involutions on a set of objects to be a queer dual equivalence, and we prove whenever such a family exists, the fundamental quasisymmetric generating function is Schur P-positive. In contrast with shifted dual equivalence, the queer dual equivalence involutions restrict to a dual equivalence when the queer involution is omitted. We highlight the difference between these two generalizations with a new application to the product of Schur P-functions.

With Stephanie van Willigenburg, European J. Combinatorics 103 (2022). arXiv:1905.11526

Abstract: Young’s lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young’s lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new poset. Using weak dual equivalence, we give a nonnegative weak composition Littlewood–Richardson rule for the key expansion of skew key polynomials, generalizing the flagged Littlewood–Richardson rule of Reiner and Shimozono.

With Nicolle S. Gonzalez, Algebraic Combinatorics 4.5 (2021) pp. 777–793. arXiv:2002.04141

Abstract: We give a crystal-theoretic proof that nonsymmetric Macdonald polynomials specialized to $t=0$ are affine Demazure characters. We explicitly construct an affine Demazure crystal on semistandard key tabloids such that removing the affine edges recovers the finite Demazure crystals constructed earlier by the authors. We also realize the filtration on highest weight modules by Demazure modules by defining explicit embedding operators which, at the level of characters, parallels the recursion operators of Knop and Sahi for specialized nonsymmetric Macdonald polynomials. Thus we prove combinatorially in type A that every affine Demazure module admits a finite Demazure flag.

Annals of Combinatorics 26.3 (2022), pp. 571–591. arXiv:1702.04051

Abstract: We use dual equivalence to give a short, combinatorial proof that Stanley symmetric functions are Schur positive. We introduce weak dual equivalence, and use it to give a short, combinatorial proof that Schubert polynomials are key positive. To demonstrate further the utility of this new tool, we use weak dual equivalence to prove a nonnegative Littlewood–Richardson rule for the key expansion of the product of a key polynomial and a Schur polynomial, and to introduce skew key polynomials that, when skewed by a partition, expand nonnegatively in the key basis.

With Nicolle S. Gonzalez, Journal of Combinatorial Theory, Series A, 182 (2021). arXiv:1901.07520

Abstract: We give an explicit, nonnegative formula for the expansion of nonsymmetric Macdonald polynomials specialized at $t=0$ in terms of Demazure characters. Our formula results from constructing Demazure crystals whose characters are the nonsymmetric Macdonald polynomials, which also gives a new proof that these specialized nonsymmetric Macdonald polynomials are positive graded sums of Demazure characters. Demazure crystals are certain truncations of classical crystals that give a combinatorial skeleton for Demazure modules. To prove our construction, we develop further properties of Demazure crystals, including an efficient algorithm for computing their characters from highest weight elements. As a corollary, we obtain a new formula for the Schur expansion of Hall–Littlewood polynomials in terms of a simple statistic on highest weight elements of our crystals.

Algebraic Combinatorics, Volume 4 (2021) no. 2, p. 359–385. arXiv:1903.0580

Abstract: Edelman and Greene generalized the Robinson–Schensted–Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman–Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial variation, but here we demonstrate how to extract explicit formulas from that machinery which may be applied to other positivity problems as well.

With Dominic Searles, Experimental Mathematics 31.1 (2022), pp 93–119. arXiv:1711.09498

Abstract: We associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert’s algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts, with each choice giving rise to a combinatorially-defined basis of polynomials. These \emph{Kohnert bases} provide a simultaneous generalization of Schubert polynomials and Demazure characters for the general linear group. Using the monomial and fundamental slide bases defined earlier by the authors, we show that Kohnert polynomials stabilize to quasisymmetric functions that are nonnegative on the fundamental basis for quasisymmetric functions. For initial applications, we define and study two new Kohnert bases. The elements of one basis are conjecturally Schubert-positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains the Schur functions.

With Ezgi Kantarci Oguz, Annals of Combinatorics 24 (2020), no. 1, 3-46. arXiv:1803.06317

Abstract: We define operators on semistandard shifted tableaux and use Stembridge’s local characterization for regular graphs to prove they define a crystal structure. This gives a new proof that Schur P-polynomials are Schur positive. We define queer crystal operators (also called odd Kashiwara operators) to construct a connected queer crystal on semistandard shifted tableaux of a given shape. Using the tensor rule for queer crystals, this provides a new proof that products of Schur P-polynomials are Schur P-positive. Finally, to facilitate applications of queer crystals in the context of Schur P-positivity, we give local axioms for queer regular graphs, generalizing Stembridge’s axioms, that partially characterize queer crystals.

With Nantel Bergeron, European J. Combin. 86 (2020). arXiv:1904.06630

Abstract: We introduce the theory of $(\mathcal{P},\rho)$-partitions, depending on a poset $\mathcal{P}$ and a map $\rho$ from $\mathcal{P}$ to positive integers. The generating function $\mathfrak{F}_{\mathcal{P},\rho}$ of $(\mathcal{P},\rho)$-partitions is a polynomial that, when the images of $\rho$ tend to infinity, tends to Stanley’s generating function of $\mathcal{P}$-partitions. Analogous to Stanley’s fundamental theorem for $\mathcal{P}$-partitions, we show the set of $(\mathcal{P},\rho)$-partitions decomposes as a disjoint union of $(\mathcal{L},\rho)$-partitions where $\mathcal{L}$ runs over the set of linear extensions of $\mathcal{P}$. In this more general context, the set of all $\mathfrak{F}_{\mathcal{L},\rho}$ for linear orders $\mathcal{L}$ over determines a basis of polynomials. We thus introduce the notion of flagged $(\mathcal{P},\rho)$-partitions, and we prove the set of all $\mathfrak{F}_{\mathcal{L},\rho}$ for flagged $(\mathcal{L},\rho)$-partitions for linear orders $\mathcal{L}$ is precisely the fundamental slide basis of the polynomial ring, introduced by the first author and Searles. Our main theorem shows that any generating function $\mathfrak{F}_{\mathcal{P},\rho}$ of flagged $(\mathcal{P},\rho)$-partitions is a positive integer linear combination g slide polynomials. As applications, we give a new proof of positivity of the slide product and, motivating our nomenclature, we also prove flagged Schur functions are slide positive.

With David Speyer, Proc. Amer. Math. Soc. 148 (2020), no. 3, 1015–1029. arXiv:1809.10125

Abstract: Schur modules give the irreducible polynomial representations of the general linear group GL_t. Viewing the symmetric group S_t as a subgroup of GL_t, we may restrict Schur modules to S_t and decompose the result into a direct sum of Specht modules, the irreducible representations of S_t. We give an equivariant Mobius inversion formula that we use to invert this expansion in the representation ring for S_t for t large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with signs alternating by degree into the Schur basis.

With Dominic Searles, Séminaire Lotharingien de Combinatoire 82B(2019), Article #31, 12pp.

Abstract: Using Kohnert’s algorithm, we associate a polynomial to any cell diagram in the positive quadrant, simultaneously generalizing Schubert polynomials and \(GL_n\) Demazure characters. We survey properties of these Kohnert polynomials and their stable limits, which are quasisymmetric functions. As a first application, we introduce and study two new bases of Kohnert polynomials, one of which stabilizes to the skew-Schur functions and is conjecturally Schubert-positive, the other stabilizes to a new basis of quasisymmetric functions that contains the Schur functions.

With Stephanie van Willigenburg, Séminaire Lotharingien de Combinatoire 82B(2019), Article #77, 12pp.

Abstract: We generalize Young’s lattice on integer partitions to a new partial order on weak compositions called the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomial basis for the polynomial ring, a generalization of the Schur basis for symmetric functions. Generalizing skew Schur functions, we define skew key polynomials in terms of the poset, and, using weak dual equivalence, we give a nonnegative composition Littlewood–Richardson rule for the key expansion of skew key polynomials.

Advances in Applied Mathematics, 107 (2019), 1-21. arXiv:1808.01281

Abstract: We study the graph on reduced words with edges given by the Coxeter relations for the symmetric group. We define a statistic on reduced words for a given permutation, analogous to Coxeter length for permutations, for which the graph becomes ranked with unique maximal element. We show this statistic extends naturally to balanced labellings, and use it to recover enumerative results of Edelman and Greene and of Reiner and Roichman.

Transactions of the American Mathematical Society, 370 (2018) no. 12, 8777–8796. arXiv:1703.02466

With Ezgi Kantarci Oguz, Séminaire Lotharingien de Combinatoire, 80B (2018) Article #26, 12pp. arXiv:1802.07352

Abstract: We define crystal operators on semistandard shifted tableaux, giving a new proof that Schur \(P\)-functions are Schur positive. We define a queer crystal operator to construct a connected queer crystal on semistandard shifted tableaux of a given shape, providing a new proof that products of Schur \(P\)-functions are Schur \(P\)-positive. We also give a rectification map from shifted tableaux to Young tableaux that commutes with the crystal operators and provides a dual algorithm to shifted insertion.

With Danjoseph Quijada, Séminaire Lotharingien de Combinatoire, 80B(2018) Article #78, 12pp.

With Nicolle S. Gonzalez, Séminaire Lotharingien de Combinatoire, 80B(2018) Article #81, 12pp.

Abstract: We construct a Demazure crystal for nonsymmetric Macdonald polynomials specialized at \(t=0\), giving a new proof that these specialized nonsymmetric Macdonald polynomials are graded sums of Demazure characters.

Electronic Journal of Combinatorics 25 (2018), no. 2, Research Paper 44. arXiv:1703.07457

With Anne Schilling, Algebraic Combinatorics, Volume 1 (2018) no. 2, p. 225–247. arXiv:1705.09649

Abstract: Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by Morse and the second author in connection with Stanley symmetric functions.

With Dominic Searles, Journal of Combinatorial Theory, Series A 156 (2018), 85–118. arXiv:1609.03507

Abstract: We introduce the quasi-key basis of the polynomial ring. We prove this basis contains the quasi-Schur polynomials of of Haglund, Luoto, Mason and van Willigenburg and that stable limits of quasi-key polynomials are quasi-Schur functions, thus giving a lifting of the quasi-Schur basis of quasisymmetric polynomials to the full polynomial ring. We introduce the combinatorial model of Kohnert tableaux, and use this model to prove that key polynomials expand positively in quasi-key polynomials which in turn expand positively in the fundamental slide polynomials introduced earlier by the authors. We give simple combinatorial formulas for these expansions in terms of Kohnert tableaux, lifting the parallel expansions of a Schur function into quasi-Schur functions into fundamental quasisymmetric functions. We further utilize Kohnert tableaux to find the precise point at which the fundamental slide expansion of a key polynomial stabilizes.

J. Comb. 9 (2018), no. 2, 279–308. arXiv:1402.2570

Abstract: By considering type B analogs of permutations and tableaux, we extend abstract dual equivalence to type B in two directions. In one direction, we define involutions on shifted tableaux that give a dual equivalence, thereby giving a new combinatorial proof of the Schur positivity of Schur Q- and P-functions. In another direction, we define an abstract shifted dual equivalence parallel to dual equivalence and prove that it can be used to establish Schur P-positivity of a function expressed as a sum of shifted fundamental quasisymmetric functions. As a first application, we give a new combinatorial proof that the product of Schur P-functions is Schur P-positive.

Séminaire Lotharingien de Combinatoire 78B(2017), Article #12, 12pp.

Abstract: We introduce several new (and recall one old) tableau models for Schubert polynomials. Applications include a bijective proof of Kohnert’s rule.

With Dominic Searles, Séminaire Lotharingien de Combinatoire 78B(2017), Article #11, 12pp.

Abstract: The fundamental slide basis of polynomials was recently introduced by the authors. We survey positivity properties of this basis, and applications to the important Schubert and key bases of polynomials.

With Dominic Searles, Adv. in Math. 306 (2017), 89–122. arXiv:1603.09744

Abstract: We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions. We also give combinatorial rules for the positive structure constants of these bases that generalize the quasi-shuffle product and shuffle product, respectively. We use this to give a Littlewood–Richardson rule for expanding a product of Schubert polynomials into fundamental slide polynomials and to give formulas for products of Stanley symmetric functions in terms of Schubert structure constants.

Forum Math. Sigma 3 (2015), e12, 33 pp. arXiv:1506.03798

Abstract: We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for establishing the symmetry and Schur positivity of quasisymmetric functions.

With Noah Forman and Jim Pitman, Electron. J. Probab. 20 (2015), no. 90, 39 pp. arXiv:1307.4967

Abstract: We examine a new path transform on 1-dimensional simple random walks and Brownian motion, the quantile transform. This transformation relates to identities in fluctuation theory due to Wendel, Port, Dassios and others, and to discrete and Brownian versions of Tanaka’s formula. For an \(n\)-step random walk, the quantile transform reorders increments according to the value of the walk at the start of each increment. We describe the distribution of the quantile transform of a simple random walk of \(n\) steps, using a bijection to characterize the number of pre-images of each possible transformed path. We deduce, both for simple random walks and for Brownian motion, that the quantile transform has the same distribution as Vervaat’s transform. For Brownian motion, the quantile transforms of the embedded simple random walks converge to a time change of the local time profile. We characterize the distribution of the local time profile, giving rise to an identity that generalizes a variant of Jeulin’s description of the local time profile of a Brownian bridge or excursion.

With Sara Billey, J. Comb. 3 (2012), no. 3, 343–399. arXiv:1201.2128

Abstract: The \(k\)-Schur functions were first introduced by Lapointe, Lascoux and Morse in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for \(k\)-Schur functions was given by Lam, Lapointe, Morse, and Shimozono as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type \(A\) and at \(t=1\) it is equivalent to the \(k\)-tableaux characterization of Lapointe and Morse. In this paper, we extend Haiman’s dual equivalence relation on standard Young tableaux to all starred strong tableaux. The elementary equivalence relations can be interpreted as labeled edges in a graph which share many of the properties of Assaf’s dual equivalence graphs. These graphs display much of the complexity of working with \(k\)-Schur functions and the interval structure on \(\widetilde{S}_{n}/S_{n}\). We introduce the notions of flattening and squashing skew starred strong tableaux in analogy with jeu de taquin slides in order to give a method to find all isomorphism types for affine dual equivalence graphs of rank 4. Finally, we state some open problems on other ways to generalize dual equivalence.

With Persi Diaconis and K. Soundararajan, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 445–456, Discrete Math. Theor. Comput. Sci. Proc., AR, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012. arXiv:1112.2650

Abstract: The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that the cards are initially cut `about in half’ and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and L-infinity distances. This analysis is possible due to the close connection between shuffling and quasisymmetric functions along with some complex analysis of a generating function.

With Peter McNamara, Journal of Combinatorial Theory, Series A 118 (2011), 277–290. arXiv:0908.0345

Abstract: The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape.

With Persi Diaconis and K. Soundararajan, Ann. Appl. Probab. 21 (2011), no. 3, 843–875. arXiv:0908.3462

Abstract: We study how many riffle shuffles are required to mix \(n\) cards if only certain features of the deck are of interest, e.g. suits disregarded or only the colors of interest. For these features, the number of shuffles drops from \(\frac{3}{2}\log_2 n\) to \(\log_2 n\). We derive closed formulae and an asymptotic `rule of thumb’ formula which is remarkably accurate.

Electronic Journal of Combinatorics 17 (2010), no. 1, Research Paper 163, 14 pp. arXiv:1002.3138

Abstract: A classic problem in enumerative combinatorics is to count the number of derangements, that is, permutations with no fixed point. Inspired by a recent generalization to facet derangements of the hypercube by Gordon and McMahon, we generalize this problem to enumerating derangements in the wreath product of any finite cyclic group with the symmetric group. We also give \(q\)- and \((q,t)\)-analogs for cyclic derangements, generalizing results of Gessel, Brenti and Chow.

With Peter McNamara, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 133–144, Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., 2010. arXiv:0908.0345

Abstract: The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case.

With Adriano Garsia, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 103–114, Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., 2009. arXiv:0905.2333

Abstract: In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module \(R_{\mu}\) is \(n!\), and they showed that the resolution of this conjecture implies the Macdonald Positivity Conjecture. Haiman proved these conjectures in 2001 using algebraic geometry, but the question remains to find an explicit basis for \(R_{\mu}\) which would give a simple proof of the dimension. Using the theory of Orbit Harmonics developed by Garsia and Haiman, we present a ”kicking basis” for \(R_{\mu}\) when \(\mu\) has two columns.

With Persi Diaconis and K. Soundararajan, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 89–102, Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., 2009. arXiv:0905.4698

Abstract: We study the Gilbert-Shannon-Reeds model for riffle shuffles and ask ’How many times must a deck of cards be shuffled for the deck to be in close to random order?’. In 1992, Bayer and Diaconis gave a solution which gives exact and asymptotic results for all decks of practical interest, e.g. a deck of 52 cards. But what if one only cares about the colors of the cards or disregards the suits focusing solely on the ranks? More generally, how does the rate of convergence of a Markov chain change if we are interested in only certain features? Our exploration of this problem takes us through random walks on groups and their cosets, discovering along the way exact formulas leading to interesting combinatorics, an ’amazing matrix’, and new analytic methods which produce a completely general asymptotic solution that is remarkable accurate.

20th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 141–152, Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., 2008. arXiv:0804.1587

Abstract: For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight \(0\). Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the \(0\)-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.

Seminaire Lotharingien de Combinatoire 60 (2008), Art. B60c, 13 pp. (electronic). arXiv:0807.0433

Abstract: Inspired by the \(k\)-inversion statistic for LLT polynomials, we define a \(k\)-inversion number and \(k\)-descent set for words. Using these, we define a new statistic on words, called the \(k\)-major index, that interpolates between the major index and inversion number. We give a bijective proof that the \(k\)-major index is equi-distributed with the major index, generalizing a classical result of Foata and rediscovering a result of Kadell. Inspired by recent work of Haglund and Stevens, we give a partial extension of these definitions and constructions to standard Young tableaux. Finally, we give an application to Macdonald polynomials made possible through connections with LLT polynomials.

Ph.D. Thesis, UC Berkeley, 2007; recipient of the 2007 Herb Alexander Prize for outstanding dissertations in pure mathematics at UC Berkeley. PDF

Abstract: We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize such constructions and prove that the generating functions of these graphs are Schur positive. We construct a graph on k-ribbon tableaux which we conjecture to be a dual equivalence graph, and we prove the conjecture for \(k \geq 3\). This implies the Schur positivity of the k-ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. From Haglund’s formula for the transformed Macdonald polynomials, this has the further consequence of a combinatorial expansion of the Macdonald-Kostka polynomials indexed by a partition with at most \(3\) columns.