Jonah Blasiak
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
Jonah Blasiak has a total of 24 co-authors. Here are the top co-authors by number of shared publications:
- Jennifer Morse
Dept. of Mathematics, University of Virginia, Charlottesville, VA
12 shared publications - Mark Haiman
Dept. of Mathematics, University of California, Berkeley, CA
7 shared publications - George H. Seelinger
Dept. of Mathematics, University of Michigan, Ann Arbor, MI
6 shared publications - Anna Pun
Baruch College, New York, NY, USA
5 shared publications - Anna Pun
Department of Mathematics, Baruch College (CUNY), New York, USA
4 shared publications - Sergey Fomin
Department of Mathematics, University of Michigan, Ann Arbor, USA
4 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
- 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
- 24 January 2025
- Published by Springer Nature in Selecta Mathematica
- Vol. 31 (1) , 1-56
Noncommutative Schur functions for posets
The machinery of noncommutative Schur functions is a general approach to Schur positivity of symmetric functions initiated by Fomin-Greene. Hwang recently adapted this theory to posets to give a new approach to the Stanley-Stembridge conjecture. We further develop this theory to prove that the symmetric function associated to any P-Knuth equivalence graph is Schur positive. This settles a conjecture of Kim and the third author, and refines results of Gasharov, Shareshian-Wachs, and Hwang on the Schur positivity of chromatic symmetric functions.
- 24 January 2025
- Published by Springer Nature in Selecta Mathematica
- Vol. 31 (1) , 1-56
Open AccessNoncommutative Schur functions for posets
The machinery of noncommutative Schur functions is a general approach to Schur positivity of symmetric functions initiated by Fomin-Greene. Hwang recently adapted this theory to posets to give a new approach to the Stanley-Stembridge conjecture. We further develop this theory to prove that the symmetric function associated to any P-Knuth equivalence graph is Schur positive. This settles a conjecture of Kim and the third author, and refines results of Gasharov, Shareshian-Wachs, and Hwang on the Schur positivity of chromatic symmetric functions.
- 23 April 2024
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
- Vol. 2024 (811) , 93-133
LLT polynomials in the Schiffmann algebra
We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one.- 23 April 2024
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
- Vol. 2024 (811) , 93-133
LLT polynomials in the Schiffmann algebra
We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one.- Open Access
- 8 March 2024
- Published by Springer Nature in Inventiones Mathematicae
- Vol. 236 (2) , 483-547
Demazure crystals and the Schur positivity of Catalan functions
Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$ k -Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of $U_{q}(\widehat{\mathfrak{sl}}_{\ell })$ U q ( sl ˆ ℓ ) -generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.
- 8 March 2024
- Published by Springer Nature in Inventiones Mathematicae
- Vol. 236 (2) , 483-547
Open AccessDemazure crystals and the Schur positivity of Catalan functions
Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$ k -Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of $U_{q}(\widehat{\mathfrak{sl}}_{\ell })$ U q ( sl ˆ ℓ ) -generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.
- 18 July 2023
- Published by Elsevier in Finite Fields and Their Applications
- Vol. 91, 102262
Multislant matrices and Jacobi–Trudi determinants over finite fields
- 18 July 2023
- Published by Elsevier in Finite Fields and Their Applications
- Vol. 91, 102262
Multislant matrices and Jacobi–Trudi determinants over finite fields
- 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
- 6 March 2023
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 11, e8
A Shuffle Theorem for Paths Under Any Line – Corrigendum
In the initial publication of this article, the 2020 Mathematics subject classification codes were listed incorrectly as 'Primary-35P15; Secondary-58C40, 53A07'. This has since been corrected in the original article.
- 6 March 2023
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 11, e8
Open AccessA Shuffle Theorem for Paths Under Any Line – Corrigendum
In the initial publication of this article, the 2020 Mathematics subject classification codes were listed incorrectly as 'Primary-35P15; Secondary-58C40, 53A07'. This has since been corrected in the original article.
- Open Access
- 6 March 2023
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 11, e7
A Proof of the Extended Delta Conjecture – Corrigendum
In the initial publication of this article, the 2020 Mathematics subject classification codes were listed incorrectly as 'Primary-35P15; Secondary-58C40, 53A07'. This has since been corrected in the original article.
- 6 March 2023
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 11, e7
Open AccessA Proof of the Extended Delta Conjecture – Corrigendum
In the initial publication of this article, the 2020 Mathematics subject classification codes were listed incorrectly as 'Primary-35P15; Secondary-58C40, 53A07'. This has since been corrected in the original article.
- 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.