arXiv subjects

Jonah Blasiak

Publications and source records attributed to Jonah Blasiak.

At least 37 records · Page 2Linked to original sources

arXiv2016

What makes a D_0 graph Schur positive?

We define a D_0 graph to be a graph whose vertex set is a subset of permutations of n, with edges of the form ...bac... <--> ...bca... or ...acb... <--> ...cab... (Knuth transformations), or ...bac... <--> ...acb... or ...bca... <--> ...cab... (rotation transformations), such that whenever the Knuth and rotation transformations at positions i-1, i, i+1 are available at a vertex, exactly one of these is an edge. The generating function of such a graph is the sum of the quasisymmetric functions associated to the descent sets of its vertices. Assaf studied D_0 graphs in the paper *Dual equivalence and Schur positivity* and showed that they provide a rich source of examples of the D graphs defined in the paper *Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity*. A key construction of Assaf expresses the coefficient of q^t in an LLT polynomial as the generating function of a certain D_0 graph. LLT polynomials are known to be Schur positive by work of Grojnowski-Haiman, and experimentation shows that many D_0 graphs have Schur positive generating functions, which suggests a vast generalization of LLT positivity in this setting. As part of a series of papers, we study D_0 graphs using the Fomin-Greene theory of noncommutative Schur functions. We construct a D_0 graph whose generating function is not Schur positive by solving a linear program related to a certain noncommutative Schur function. We go on to construct a D graph on the same vertex set as this D_0 graph.

math.CO↗

arXiv2016

Rules of Three for commutation relations

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 enough that these relations hold for subsets of size one, two, and three.

math.RA↗

arXiv2016

Noncommutative Schur functions, switchboards, and Schur positivity

The machinery of noncommutative Schur functions provides a general tool for obtaining Schur expansions for combinatorially defined symmetric functions. We extend this approach to a wider class of symmetric functions, explore its strengths and limitations, and obtain new results on Schur positivity. We introduce combinatorial gadgets called switchboards, an adaptation of the D graphs of S. Assaf, and show how symmetric functions associated to them (which include LLT, Macdonald, Stanley, and stable Grothendieck polynomials) fit into the noncommutative Schur functions approach. This extends earlier work by T. Lam, and by C. Greene and the second author, and provides new tools for obtaining combinatorial formulas for Schur expansions of LLT polynomials. This paper can be regarded as a "prequel" to (and, partly, a review of) arXiv:1411.3624, arXiv:1411.3646, and arXiv:1510.00644.

math.CO↗

arXiv2015

Kronecker coefficients and noncommutative super Schur functions

The theory of noncommutative Schur functions can be used to obtain positive combinatorial formulae for the Schur expansion of various classes of symmetric functions, as shown by Fomin and Greene. We develop a theory of noncommutative super Schur functions and use it to prove a positive combinatorial rule for the Kronecker coefficients where one of the partitions is a hook, recovering previous results of the two authors. This method also gives a precise connection between this rule and a heuristic for Kronecker coefficients first investigated by Lascoux.

math.CO↗

arXiv2014

Haglund's conjecture on 3-column Macdonald polynomials

We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining this result with the expression of Haglund, Haiman, and Loehr for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.

math.CO↗

arXiv2014

Nonstandard braid relations and Chebyshev polynomials

A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product of two §_r-irreducibles into irreducibles. Mulmuley and Sohoni attempt to solve this problem using canonical basis theory, by first constructing a nonstandard Hecke algebra B_r, which, though not a Hopf algebra, is a u-analogue of the Hopf algebra \CC §_r in some sense (where u is the Hecke algebra parameter). For r=3, we study this Hopf-like structure in detail. We define a nonstandard Hecke algebra \bar{\H}^{(k)}_3 \subseteq \H_3^{\tsr k}, determine its irreducible representations over \QQ(u), and show that it has a presentation with a nonstandard braid relation that involves Chebyshev polynomials evaluated at \frac{1}{u + u^{-1}}. We generalize this to Hecke algebras of dihedral groups. We go on to show that these nonstandard Hecke algebras have bases similar to the Kazhdan-Lusztig basis of \H_3 and are cellular algebras in the sense of Graham and Lehrer.

math.RT↗

arXiv2014

Subalgebras of the Fomin-Kirillov algebra

The Fomin-Kirillov algebra $\mathcal E_n$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on $n$ vertices. For any graph $G$ on $n$ vertices, we define $\mathcal E_G$ to be the subalgebra of $\mathcal E_n$ generated by the edges of $G$. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $\mathcal E_G$ is a free $\mathcal E_H$-module for any $H\subseteq G$, and if $\mathcal E_G$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for $\mathcal E_G$ when $G$ is a simply-laced finite Dynkin diagram or a cycle, in particular showing that $\mathcal E_G$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of $\mathcal E_{\tilde{D}_n}$, $\mathcal E_{\tilde{E}_6}$, and $\mathcal E_{\tilde{E}_7}$, as well as for which graphs $G$ on six vertices $\mathcal E_G$ is finite-dimensional.

math.QA↗

arXiv2013

Quantum Schur-Weyl duality and projected canonical bases

Let \H_r be the generic type A Hecke algebra defined over \ZZ[u, u^{-1}]. The Kazhdan-Lusztig bases \{C_w\}_{w \in §_r} and \{C'_w\}_{w \in §_r} of \H_r give rise to two different bases of the Specht module M_λ, λ\vdash r, of \H_r. These bases are not equivalent and we show that the transition matrix S(λ) between the two is the identity at u = 0 and u = \infty. To prove this, we first prove a similar property for the transition matrices \tilde{T}, \tilde{T}' between the Kazhdan-Lusztig bases and their projected counterparts \{\tilde{C}_w\}_{w \in §_r}, \{\tilde{C}'_w\}_{w \in §_r}, where \tilde{C}_w := C_w p_λ, \tilde{C}'_w := C'_w p_λand p_λis the minimal central idempotent corresponding to the two-sided cell containing w. We prove this property of \tilde{T},\tilde{T}' using quantum Schur-Weyl duality and results about the upper and lower canonical basis of V^{\tsr r} (V the natural representation of U_q(\gl_n)) from \cite{GL, FKK, Brundan}. We also conjecture that the entries of S(λ) have a certain positivity property.

math.RT↗

arXiv2013

Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem

The Kronecker coefficient g_{λμν} is the multiplicity of the GL(V)\times GL(W)-irreducible V_λ\otimes W_μin the restriction of the GL(X)-irreducible X_νvia the natural map GL(V)\times GL(W) \to GL(V \otimes W), where V, W are \mathbb{C}-vector spaces and X = V \otimes W. A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. We construct two quantum objects for this problem, which we call the nonstandard quantum group and nonstandard Hecke algebra. We show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality. Using these nonstandard objects as a guide, we follow the approach of Adsul, Sohoni, and Subrahmanyam to construct, in the case dim(V) = dim(W) =2, a representation \check{X}_νof the nonstandard quantum group that specializes to Res_{GL(V) \times GL(W)} X_νat q=1. We then define a global crystal basis +HNSTC(ν) of \check{X}_νthat solves the two-row Kronecker problem: the number of highest weight elements of +HNSTC(ν) of weight (λ,μ) is the Kronecker coefficient g_{λμν}. We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the U_q(\sl_2) graphical calculus, and use this to organize the crystal components of +HNSTC(ν) into eight families. This yields a fairly simple, explicit and positive formula for two-row Kronecker coefficients, generalizing a formula of Brown, van Willigenburg, and Zabrocki. As a byproduct of the approach, we also obtain a rule for the decomposition of Res_{GL_2 \times GL_2 \rtimes §_2} X_νinto irreducibles.

cs.CC↗

arXiv2012

Kronecker coefficients for one hook shape

We give a positive combinatorial formula for the Kronecker coefficient g_{lambda mu(d) nu} for any partitions lambda, nu of n and hook shape mu(d) := (n-d,1^d). Our main tool is Haiman's \emph{mixed insertion}. This is a generalization of Schensted insertion to \emph{colored words}, words in the alphabet of barred letters \bar{1},\bar{2},... and unbarred letters 1,2,.... We define the set of \emph{colored Yamanouchi tableaux of content lambda and total color d} (CYT_{lambda, d}) to be the set of mixed insertion tableaux of colored words w with exactly d barred letters and such that w^{blft} is a Yamanouchi word of content lambda, where w^{blft} is the ordinary word formed from w by shuffling its barred letters to the left and then removing their bars. We prove that g_{lambda mu(d) nu} is equal to the number of CYT_{lambda, d} of shape nu with unbarred southwest corner.

math.CO↗

arXiv2012

Representation theory of the nonstandard Hecke algebra

The nonstandard Hecke algebra \check{\mathscr{H}}_r was defined by Mulmuley and Sohoni to study the Kronecker problem. We study a quotient \check{\mathscr{H}}_{r,2} of \check{\mathscr{H}}_r, called the nonstandard Temperley-Lieb algebra, which is a subalgebra of the symmetric square of the Temperley-Lieb algebra TL_r. We give a complete description of its irreducible representations. We find that the restriction of an \check{\mathscr{H}}_{r,2}-irreducible to \check{\mathscr{H}}_{r-1,2} is multiplicity-free, and as a consequence, any \check{\mathscr{H}}_{r,2}-irreducible has a seminormal basis that is unique up to a diagonal transformation.

math.RT↗

arXiv2010

Cyclage, catabolism, and the affine Hecke algebra

We identify a subalgebra \pH_n of the extended affine Hecke algebra \eH_n of type A. The subalgebra \pH_n is a \u-analogue of the monoid algebra of §_n \ltimes \ZZ_{\geq 0}^n and inherits a canonical basis from that of \eH_n. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term \emph{positive affine tableaux} (PAT). We then exhibit a cellular subquotient \R_{1^n} of \pH_n that is a \u-analogue of the ring of coinvariants \CC[y_1,...,y_n]/(e_1,...,e_n) with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element π\in \pH_n corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that \R_{1^n} has cellular quotients \R_λthat are \u-analogues of the Garsia-Procesi modules R_λwith left cells labeled by (a PAT version of) the λ-catabolizable tableaux. We give a conjectural description of a cellular filtration of \pH_n, the subquotients of which are isomorphic to dual versions of \R_λunder the perfect pairing on \R_{1^n}. We conjecture how this filtration relates to the combinatorics of the cells of \eH_n worked out by Shi, Lusztig, and Xi. We also conjecture that the k-atoms of Lascoux, Lapointe, and Morse and the R-catabolizable tableaux of Shimozono and Weyman have cellular counterparts in \pH_n. We extend the idea of atom copies of Lascoux, Lapoint, and Morse to positive affine tableaux and give descriptions, mostly conjectural, of some of these copies in terms of catabolizability.

math.CO↗

arXiv2009

An insertion algorithm for catabolizability

Motivated by our recent work relating canonical bases to combinatorics of Garsia-Procesi modules \cite{B}, we give an insertion algorithm that computes the catabolizability of the insertion tableau of a standard word. This allows us to characterize catabolizability as the statistic on words invariant under Knuth transformations, certain (co)rotations, and a new operation called a catabolism transformation. We also prove a Greene's Theorem-like characterization of catabolizability, and a result about how cocyclage changes catabolizability, strengthening a similar result in \cite{SW}.

math.CO↗

arXiv2009

A factorization theorem for affine Kazhdan-Lusztig basis elements

The lowest two-sided cell of the extended affine Weyl group $W_e$ is the set $\{w \in W_e: w = x \cdot w_0 \cdot z, \text{for some} x,z \in W_e\}$, denoted $W_{(ν)}$. We prove that for any $w \in W_{(ν)}$, the canonical basis element $\C_w$ can be expressed as $\frac{1}{[n]!} χ_λ({\y}) \C_{v_1 w_0} \C_{w_0 v_2}$, where $χ_λ({\y})$ is the character of the irreducible representation of highest weight $λ$ in the Bernstein generators, and $v_1$ and $v_2^{-1}$ are what we call primitive elements. Primitive elements are naturally in bijection with elements of the finite Weyl group $W_f \subseteq W_e$, thus this theorem gives an expression for any $\C_w$, $w \in W_{(ν)}$ in terms of only finitely many canonical basis elements. After completing this paper, we realized that this result was first proved by Xi in \cite{X}. The proof given here is significantly different and somewhat longer than Xi's, however our proof has the advantage of being mostly self-contained, while Xi's makes use of results of Lusztig from \cite{L Jantzen} and Cells in affine Weyl groups I-IV and the positivity of Kazhdan-Lusztig coefficients.

math.RT↗

arXiv2008

$W$-graph versions of tensoring with the $§_n$ defining representation

We further develop the theory of inducing $W$-graphs worked out by Howlett and Yin in \cite{HY1}, \cite{HY2}, focusing on the case $W = §_n$. Our main application is to give two $W$-graph versions of tensoring with the $§_n$ defining representation $V$, one being $\H \tsr_{\H_J} -$ for $\H, \H_J$ the Hecke algebras of $§_n, §_{n-1}$ and the other $(\pH \tsr_{\H} -)_1$, where $\pH$ is a subalgebra of the extended affine Hecke algebra and the subscript signifies taking the degree 1 part. We look at the corresponding $W$-graph versions of the projection $V \tsr V \tsr - \to S^2 V \tsr -$. This does not send canonical basis elements to canonical basis elements, but we show that it approximates doing so as the Hecke algebra parameter $\u \to 0$. We make this approximation combinatorially explicit by determining it on cells. Also of interest is a combinatorial conjecture stating the restriction of $\H$ to $\H_J$ is "weakly multiplicity-free" for $|J| = n-1$, and a partial determination of the map $\H \tsr_{\H_J} \H \xrightarrow{\counit} \H$ on canonical basis elements, where $\counit$ is the counit of adjunction.

math.RT↗

arXiv2005

The toric ideal of a graphic matroid is generated by quadrics

Describing minimal generating sets of toric ideals is a well-studied and difficult problem. Neil White conjectured in 1980 that the toric ideal associated to a matroid is generated by quadrics corresponding to single element symmetric exchanges. We give a combinatorial proof of White's conjecture for graphic matroids.

math.CO↗

arXiv2005

A special case of Hadwiger's conjecture

We investigate Hadwiger's conjecture for graphs with no stable set of size 3. Such a graph on at least 2t-1 vertices is not t-1 colorable, so is conjectured to have a $K_t$ minor. There is a strengthening of Hadwiger's conjecture in this case, which states that there is always a minor in which the preimages of the vertices of $K_t$ are connected subgraphs of size one or two. We prove this strengthened version for graphs whose complement has an even number of vertices and fractional chromatic number less than 3. We investigate several possible generalizations and obtain counterexamples for some and improved results from others. We also show that for sufficiently large $n=|V(G)|$, a graph with no stable set of size 3 has a $K_{1/9 n^{4/5}}$ minor using only sets of size one or two as preimages of vertices.

math.CO↗

arXiv2004

Longest common subsequences and the Bernoulli matching model: numerical work and analyses of the r-reach simplification

The expected length of longest common subsequences is a problem that has been in the literature for at least twenty five years. Determining the limiting constants γ_k appears to be quite difficult, and the current best bounds leave much room for improvement. Boutet de Monvel explores an independent version of the problem he calls the Bernoulli Matching model. He explores this problem and its relation to the longest common subsequence problem. This paper continues this pursuit by focusing on a simplification we term r-reach. For the string model, L_r(u,v) is the longest common subsequence of u and v given that each matched pair of letters is no more than r letters apart.

math.PR↗