Jonah Blasiak
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View article: The nonsymmetric shuffle theorem
The nonsymmetric shuffle theorem Open
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 M…
View article: Dens, nests and the Loehr-Warrington conjecture
Dens, nests and the Loehr-Warrington conjecture Open
We prove and extend the longest-standing conjecture in ‘-Catalan combinatorics,’ namely, the combinatorial formula for conjectured by Loehr and Warrington, where is a Schur function and is an eigenoperator on Macdonald polynomials. Our …
View article: Noncommutative Schur functions for posets
Noncommutative Schur functions for posets Open
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 conje…
View article: Finite Matrix Multiplication Algorithms from Infinite Groups
Finite Matrix Multiplication Algorithms from Infinite Groups Open
The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group G satisfying a simple combinatorial condition (the Triple Product Property). …
View article: A raising operator formula for Macdonald polynomials
A raising operator formula for Macdonald polynomials Open
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 modif…
View article: Pseudorandom Bits for Non-Commutative Programs
Pseudorandom Bits for Non-Commutative Programs Open
We obtain new explicit pseudorandom generators for several computational models involving groups. Our main results are as follows: 1) We consider read-once group-products over a finite group G, i.e., tests of the form ∏_{i=1}^n (g_i)^{x_i}…
View article: LLT polynomials in the Schiffmann algebra
LLT polynomials in the Schiffmann algebra Open
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 Λ ( X m , n ) ⊂ E \Lambda(X^{m{,}n})\subset\…
View article: Demazure crystals and the Schur positivity of Catalan functions
Demazure crystals and the Schur positivity of Catalan functions Open
Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$ -Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catala…
View article: A raising operator formula for Macdonald polynomials
A raising operator formula for Macdonald polynomials Open
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 Ma…
View article: Multislant matrices and Jacobi--Trudi determinants over finite fields
Multislant matrices and Jacobi--Trudi determinants over finite fields Open
The problem of counting the $\mathbb{F}_q$-valued points of a variety has been well-studied from algebro-geometric, topological, and combinatorial perspectives. We explore a combinatorially flavored version of this problem studied by Anzis…
View article: A Shuffle Theorem for Paths Under Any Line
A Shuffle Theorem for Paths Under Any Line Open
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 sid…
View article: A Proof of the Extended Delta Conjecture
A Proof of the Extended Delta Conjecture Open
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 sy…
View article: Noncommutative Schur functions for posets
Noncommutative Schur functions for posets Open
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 conje…
View article: Matrix multiplication via matrix groups
Matrix multiplication via matrix groups Open
In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining $ω= 2$, while other families of …
View article: A proof of the Extended Delta Conjecture
A proof of the Extended Delta Conjecture Open
We prove the Extended Delta Conjecture of Haglund, Remmel, and Wilson, a combinatorial formula for $Δ_{h_l}Δ' _{e_k} e_{n}$, where $Δ' _{e_k}$ and $Δ_{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We act…
View article: A Shuffle Theorem for Paths Under Any Line
A Shuffle Theorem for Paths Under Any Line Open
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 si…
View article: $K$-theoretic Catalan functions
$K$-theoretic Catalan functions Open
We prove that the $K$-$k$-Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. Lam-Schilli…
View article: Demazure crystals and the Schur positivity of Catalan functions
Demazure crystals and the Schur positivity of Catalan functions Open
Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan f…
View article: The DARK side of generalized Demazure crystals
The DARK side of generalized Demazure crystals Open
Naoi showed that tensor products of perfect Kirillov-Reshetikhin crystals are isomorphic to certain generalized Demazure crystals. We extend Naoi's results to address distinguished subsets of these tensor products. In type A, these are nat…
View article: k-Schur expansions of Catalan functions
k-Schur expansions of Catalan functions Open
View article: Catalan functions and 𝑘-Schur positivity
Catalan functions and 𝑘-Schur positivity Open
We prove that graded-Schur functions are-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded-Schur functions and …
View article: Expansions of Catalan Functions
Expansions of Catalan Functions Open
The k-Schur functions are objects which have been studied extensively by many authors. This thesis discusses a new method of studying these functions through a larger class of symmetric functions, the Catalan Functions. We discuss general …
View article: Catalan functions and $k$-Schur positivity
Catalan functions and $k$-Schur positivity Open
We prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur fu…
View article: Kronecker coefficients and noncommutative super Schur functions
Kronecker coefficients and noncommutative super Schur functions Open
View article: On cap sets and the group-theoretic approach to matrix multiplication
On cap sets and the group-theoretic approach to matrix multiplication Open
In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent ω of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed s…
View article: Noncommutative Schur functions, switchboards, and Schur positivity
Noncommutative Schur functions, switchboards, and Schur positivity Open
View article: Rules of Three for commutation relations
Rules of Three for commutation relations Open
We investigate the following surprisingly widespread phenomenon which we call The Rule of Three: in order for a particular kind of commutation relation to hold for subsequences of elements of a ring labeled by any subset of indices, it is …