scholarly journals Dual equivalence graphs revisited and the explicit Schur expansion of a family of LLT polynomials

2013 ◽  
Vol 39 (2) ◽  
pp. 389-428 ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Llt Polynomials ◽  
Dual Equivalence ◽  
Dual Equivalence Graphs

scholarly journals Coxeter-Knuth Graphs and a Signed Little Map for Type B Reduced Words

10.37236/4384 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Sara Billey ◽  
Zachary Hamaker ◽  
Austin Roberts ◽  
Benjamin Young
Keyword(s):  
Structure Theorem ◽  
Type A ◽  
Axiomatic Characterization ◽  
Schubert Calculus ◽  
Type B ◽  
Dual Equivalence ◽  
Carry Over ◽  
Dual Equivalence Graphs ◽  
Reduced Words

We define an analog of David Little’s algorithm for reduced words in type B, and investigate its main properties. In particular, we show that our algorithm preserves the recording tableau of Kraśkiewicz insertion, and that it provides a bijective realization of the Type B transition equations in Schubert calculus. Many other aspects of type A theory carry over to this new setting. Our primary tool is a shifted version of the dual equivalence graphs defined by Assaf and further developed by Roberts. We provide an axiomatic characterization of shifted dual equivalence graphs, and use them to prove a structure theorem for the graph of Type B Coxeter-Knuth relations. 

scholarly journals DUAL EQUIVALENCE GRAPHS I: A NEW PARADIGM FOR SCHUR POSITIVITY

2015 ◽  
Vol 3 ◽  
Author(s):  
SAMI H. ASSAF
Keyword(s):  
Systematic Study ◽  
Generating Functions ◽  
Universal Method ◽  
New Paradigm ◽  
Quasisymmetric Functions ◽  
Dual Equivalence ◽  
Combinatorial Construction ◽  
Dual Equivalence Graphs ◽  
Schur Positive

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.

scholarly journals A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
S. Assaf
Keyword(s):  
Graph Structure ◽  
Young Tableaux ◽  
Polynomial Representation ◽  
Crystal Graphs ◽  
Dual Equivalence ◽  
Weyl Duality ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Do So ◽  
Dual Equivalence Graphs

International audience 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.

Dual Equivalence Graphs Revisited

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Macdonald Polynomials ◽  
Quasisymmetric Function ◽  
Schur Function ◽  
Connected Component ◽  
Slight Improvement ◽  
Quasisymmetric Functions ◽  
Dual Equivalence ◽  
International Audience ◽  
The Creation ◽  
Dual Equivalence Graphs

International audience In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions so that the sum of the weights of a connected component is a single Schur function. In this paper, we improve on Assaf's axiomatization of such graphs, giving locally testable criteria that are more easily verified by computers. We then demonstrate the utility of this result by giving explicit Schur expansions for a family of Lascoux-Leclerc-Thibon polynomials. This family properly contains the previously known case of polynomials indexed by two skew shapes, as was described in a 1995 paper by Christophe Carré and Bernard Leclerc. As an immediate corollary, we gain an explicit Schur expansion for a family of modified Macdonald polynomials in terms of Yamanouchi words. This family includes all polynomials indexed by shapes with less than four cells in the first row and strictly less than three cells in the second row, a slight improvement over the known two column case described in 2005 by James Haglund, Mark Haiman, and Nick Loehr.

scholarly journals P-Partitions and p-Positivity

2019 ◽  
Author(s):  
Per Alexandersson ◽  
Robin Sulzgruber
Keyword(s):  
Linear Combination ◽  
Generating Functions ◽  
Quasisymmetric Functions ◽  
Vertical Strip ◽  
Power Sums ◽  
Llt Polynomials ◽  
Tutte Polynomials

AbstractUsing the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse $P$-partitions expand positively into quasisymmetric power sums. Consequently, any nonnegative linear combination of such functions is $p$-positive whenever it is symmetric. As an application, we derive positivity results for chromatic quasisymmetric functions, unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials, and the more general $B$-polynomials, matroid quasisymmetric functions, and certain Eulerian quasisymmetric functions, thus reproving and improving on numerous results in the literature.

Realcompact spaces and regular σ -frames

1984 ◽  
Vol 96 (1) ◽  
pp. 73-79 ◽  
Author(s):  
Christopher R. A. Gilmour
Keyword(s):  
Topological Space ◽  
Dual Equivalence ◽  
Tychonoff Spaces

AbstractWe show that there is an adjoint dual equivalence between realcompact Alexandroff spaces and the Alexandroff σ-frames. This gives a corresponding adjoint duality for realcompact Tychonoff spaces. Consequently we characterize lattice theoretically the cozero-sets of a topological space.

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