Anna Pun
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Anna Pun has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- George H. Seelinger
Dept. of Mathematics, University of Michigan, Ann Arbor, MI
5 shared publications - Jennifer Morse
Dept. of Mathematics, University of Virginia, Charlottesville, VA
5 shared publications - Jonah Blasiak
Dept. of Mathematics, Drexel University, Philadelphia, PA
5 shared publications - Mark Haiman
Dept. of Mathematics, University of California, Berkeley, CA
4 shared publications Catherine Yan
3 shared publicationsChenchen Zhao
3 shared publications
Latest Publications
- Open Access
The nonsymmetric shuffle theorem
The shuffle conjecture of Haglund et al. expresses the symmetric function $\nabla e_n$ as a sum over labeled Dyck paths. Here $\nabla$ is an operator on symmetric functions defined in terms of its diagonal action on the basis of modified Macdonald polynomials. The shuffle conjecture was later refined by Haglund-Morse-Zabrocki to the compositional shuffle conjecture, expressing $\nabla C_α$ as a sum over labeled Dyck paths with touchpoints specified by $α$, where $C_α$ is a compositional Hall-Littlewood polynomial. Carlsson-Mellit settled both versions by developing the theory of a variant of the DAHA called the double Dyck path algebra. In a recent paper, we discovered a notion of nonsymmetric plethsym which led us to a construction of modified nonsymmetric Macdonald polynomials $\mathsf{H}_{η|λ}(\mathbf{x};q,t)$. These polynomials Weyl symmetrize to their symmetric counterparts and are conjecturally atom positive. Here we introduce a nonsymmetric version $\boldsymbol{\nabla}$ of $\nabla$, now acting diagonally on the basis given by the functions $\mathsf{H}_{η|λ}(\mathbf{x};q,t)$. Weaving together our theory with results of Carlsson-Mellit and Mellit, we establish a nonsymmetric version of the compositional shuffle theorem, which equates $\boldsymbol{\nabla}^{-1}$ applied to a nonsymmetric version $\mathsf{C}_α$ of $C_α$ with a sum over flagged labeled Dyck paths with touchpoints given by $α$. This combinatorial sum is conjecturally atom positive, refining the known Schur positivity of its symmetric counterpart.Open AccessThe nonsymmetric shuffle theorem
The shuffle conjecture of Haglund et al. expresses the symmetric function $\nabla e_n$ as a sum over labeled Dyck paths. Here $\nabla$ is an operator on symmetric functions defined in terms of its diagonal action on the basis of modified Macdonald polynomials. The shuffle conjecture was later refined by Haglund-Morse-Zabrocki to the compositional shuffle conjecture, expressing $\nabla C_α$ as a sum over labeled Dyck paths with touchpoints specified by $α$, where $C_α$ is a compositional Hall-Littlewood polynomial. Carlsson-Mellit settled both versions by developing the theory of a variant of the DAHA called the double Dyck path algebra. In a recent paper, we discovered a notion of nonsymmetric plethsym which led us to a construction of modified nonsymmetric Macdonald polynomials $\mathsf{H}_{η|λ}(\mathbf{x};q,t)$. These polynomials Weyl symmetrize to their symmetric counterparts and are conjecturally atom positive. Here we introduce a nonsymmetric version $\boldsymbol{\nabla}$ of $\nabla$, now acting diagonally on the basis given by the functions $\mathsf{H}_{η|λ}(\mathbf{x};q,t)$. Weaving together our theory with results of Carlsson-Mellit and Mellit, we establish a nonsymmetric version of the compositional shuffle theorem, which equates $\boldsymbol{\nabla}^{-1}$ applied to a nonsymmetric version $\mathsf{C}_α$ of $C_α$ with a sum over flagged labeled Dyck paths with touchpoints given by $α$. This combinatorial sum is conjecturally atom positive, refining the known Schur positivity of its symmetric counterpart. - Open Access
Minimal elements in the skew extended 0-Hecke poset
The row-strict 0-Hecke action on standard immaculate skew tableaux was studied by the present authors, who showed that it gives rise to a bounded poset, called the \emph{skew immaculate Hecke poset}, and consequently to a cyclic 0-Hecke module. It was further shown that the subposet of skew standard extended immaculate tableaux always has a unique maximal element, but may have multiple minimal elements. In this paper we focus on these minimal elements, completely classifying them for a family of skew shapes that we call \emph{lobsters}. Moreover, we prove that when the skew shape is connected, the skew extended Hecke poset does have a unique minimal element, thereby showing that the associated 0-Hecke module is cyclic for both the row-strict and the dual immaculate actions.
Open AccessMinimal elements in the skew extended 0-Hecke poset
The row-strict 0-Hecke action on standard immaculate skew tableaux was studied by the present authors, who showed that it gives rise to a bounded poset, called the \emph{skew immaculate Hecke poset}, and consequently to a cyclic 0-Hecke module. It was further shown that the subposet of skew standard extended immaculate tableaux always has a unique maximal element, but may have multiple minimal elements. In this paper we focus on these minimal elements, completely classifying them for a family of skew shapes that we call \emph{lobsters}. Moreover, we prove that when the skew shape is connected, the skew extended Hecke poset does have a unique minimal element, thereby showing that the associated 0-Hecke module is cyclic for both the row-strict and the dual immaculate actions.
- Open Access
- 11 April 2025
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 32 (2) , P2.11
The Skew Immaculate Hecke Poset and 0-Hecke Modules
The immaculate Hecke poset was introduced and investigated by Niese, Sundaram, van Willigenburg, Vega and Wang, who established the full poset structure, and determined modules for the 0-Hecke algebra action on immaculate and row-strict immaculate tableaux. In this paper, we extend their results by introducing the skew immaculate Hecke poset. We investigate the poset structure, and construct modules for the 0-Hecke algebra action on skew immaculate and skew row-strict immaculate tableaux, thus showing that the skew immaculate Hecke poset captures representation-theoretic information analogous to the immaculate Hecke poset. We also describe branching rules for the resulting skew modules.
- 11 April 2025
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 32 (2) , P2.11
Open AccessThe Skew Immaculate Hecke Poset and 0-Hecke Modules
The immaculate Hecke poset was introduced and investigated by Niese, Sundaram, van Willigenburg, Vega and Wang, who established the full poset structure, and determined modules for the 0-Hecke algebra action on immaculate and row-strict immaculate tableaux. In this paper, we extend their results by introducing the skew immaculate Hecke poset. We investigate the poset structure, and construct modules for the 0-Hecke algebra action on skew immaculate and skew row-strict immaculate tableaux, thus showing that the skew immaculate Hecke poset captures representation-theoretic information analogous to the immaculate Hecke poset. We also describe branching rules for the resulting skew modules.
- Open Access
- 19 February 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13, e47
A raising operator formula for Macdonald polynomials
We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde {H}_{\mu }(X;q,t)$ , which follows from our recent formula for $\nabla $ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions $\tilde {H}^{1,n}(X;q,t)$ that we call $1,n$ -Macdonald polynomials, which reduce to a scalar multiple of $\tilde {H}_{\mu }(X;q,t)$ when $n=1$ . We conjecture that the coefficients of $1,n$ -Macdonald polynomials in terms of Schur functions belong to ${\mathbb N}[q,t]$ , generalizing Macdonald positivity.- 19 February 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13, e47
Open AccessA raising operator formula for Macdonald polynomials
We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde {H}_{\mu }(X;q,t)$ , which follows from our recent formula for $\nabla $ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions $\tilde {H}^{1,n}(X;q,t)$ that we call $1,n$ -Macdonald polynomials, which reduce to a scalar multiple of $\tilde {H}_{\mu }(X;q,t)$ when $n=1$ . We conjecture that the coefficients of $1,n$ -Macdonald polynomials in terms of Schur functions belong to ${\mathbb N}[q,t]$ , generalizing Macdonald positivity. - Open Access
On the Correspondence Between Integer Sequences and Vacillating Tableaux
A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_λ f^λm_k^λ$ for $n \geq 2k$, where $λ$ ranges over integer partitions of $n$, $f^λ$ is the number of standard Young tableaux of shape $λ$, and $m_k^λ$ is the number of vacillating tableaux of shape $λ$ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair of tableaux of the same shape, where one is a standard Young tableau and the other is a vacillating tableau. In this paper, we study the fine properties of Halverson and Lewandowski's bijection and explore the correspondence between integer sequences and the vacillating tableaux via the map $DI_n^k$ for general integers $n$ and $k$. In particular, we characterize the integer sequences $\boldsymbol{i}$ whose corresponding shape, $λ$, in the image $DI_n^k(\boldsymbol{i})$, satisfies $λ_1 = n$ or $λ_1 = n-k$.Open AccessOn the Correspondence Between Integer Sequences and Vacillating Tableaux
A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_λ f^λm_k^λ$ for $n \geq 2k$, where $λ$ ranges over integer partitions of $n$, $f^λ$ is the number of standard Young tableaux of shape $λ$, and $m_k^λ$ is the number of vacillating tableaux of shape $λ$ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair of tableaux of the same shape, where one is a standard Young tableau and the other is a vacillating tableau. In this paper, we study the fine properties of Halverson and Lewandowski's bijection and explore the correspondence between integer sequences and the vacillating tableaux via the map $DI_n^k$ for general integers $n$ and $k$. In particular, we characterize the integer sequences $\boldsymbol{i}$ whose corresponding shape, $λ$, in the image $DI_n^k(\boldsymbol{i})$, satisfies $λ_1 = n$ or $λ_1 = n-k$. - Open Access
On the Limiting Vacillating Tableaux for Integer Sequences
A fundamental identity in the representation theory of the partition algeba is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape $\lambda$ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair consisting of a standard Young tableau and a vacillating tableau. In this paper, we show that for a given integer sequence $\boldsymbol{i}$, when $n$ is sufficiently large, the vacillating tableaux determined by $DI_n^k(\boldsymbol{i})$ become stable when $n \rightarrow \infty$; the limit is called the limiting vacillating tableau for $\boldsymbol{i}$. We give a characterization of the set of limiting vacillating tableaux and presents explicit formulas that enumerate those vacillating tableaux.Open AccessOn the Limiting Vacillating Tableaux for Integer Sequences
A fundamental identity in the representation theory of the partition algeba is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape $\lambda$ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair consisting of a standard Young tableau and a vacillating tableau. In this paper, we show that for a given integer sequence $\boldsymbol{i}$, when $n$ is sufficiently large, the vacillating tableaux determined by $DI_n^k(\boldsymbol{i})$ become stable when $n \rightarrow \infty$; the limit is called the limiting vacillating tableau for $\boldsymbol{i}$. We give a characterization of the set of limiting vacillating tableaux and presents explicit formulas that enumerate those vacillating tableaux. - Open Access
Combinatorial Identities for Vacillating Tableaux
Vacillating tableaux are sequences of integer partitions that satisfy specific conditions. The concept of vacillating tableaux stems from the representation theory of the partition algebra and the combinatorial theory of crossings and nestings of matchings and set partitions. In this paper, we further investigate the enumeration of vacillating tableaux and derive multiple combinatorial identities and integer sequences relating to the number of vacillating tableaux, simplified vacillating tableaux, and limiting vacillating tableaux.
Open AccessCombinatorial Identities for Vacillating Tableaux
Vacillating tableaux are sequences of integer partitions that satisfy specific conditions. The concept of vacillating tableaux stems from the representation theory of the partition algebra and the combinatorial theory of crossings and nestings of matchings and set partitions. In this paper, we further investigate the enumeration of vacillating tableaux and derive multiple combinatorial identities and integer sequences relating to the number of vacillating tableaux, simplified vacillating tableaux, and limiting vacillating tableaux.
- Open Access
A raising operator formula for Macdonald polynomials
We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde{H}_{μ}(X;q,t)$, which follows from our recent formula for $\nabla$ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions $\tilde{H}^{1,n}(X;q,t)$ that we call $1,n$-Macdonald polynomials, which reduce to a scalar multiple of $\tilde{H}_μ(X;q,t)$ when $n=1$. We conjecture that the coefficients of $1,n$-Macdonald polynomials in terms of Schur functions belong to $\mathbb{N}[q,t]$, generalizing Macdonald positivity.Open AccessA raising operator formula for Macdonald polynomials
We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde{H}_{μ}(X;q,t)$, which follows from our recent formula for $\nabla$ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions $\tilde{H}^{1,n}(X;q,t)$ that we call $1,n$-Macdonald polynomials, which reduce to a scalar multiple of $\tilde{H}_μ(X;q,t)$ when $n=1$. We conjecture that the coefficients of $1,n$-Macdonald polynomials in terms of Schur functions belong to $\mathbb{N}[q,t]$, generalizing Macdonald positivity. - Open Access
- 22 February 2023
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 11
A Shuffle Theorem for Paths Under Any Line
We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whosexandyintercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $\operatorname {\mathrm {GL}}_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.- 22 February 2023
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 11
Open AccessA Shuffle Theorem for Paths Under Any Line
We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whosexandyintercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $\operatorname {\mathrm {GL}}_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials. - Top Cited PapersOpen Access
- 22 February 2023
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 11, e6
A Proof of the Extended Delta Conjecture
We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$ , where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.- 22 February 2023
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 11, e6
Top Cited PapersOpen AccessA Proof of the Extended Delta Conjecture
We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$ , where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.