#Dictionary
This page is a quick glossary for terms that appear across the site. Most entries point to a fuller page where the term is used in context.
#Algebra and representation theory
Character. The character of a representation is the trace of each group element acting on the representation space. For \(\symS_n,\) the Frobenius characteristic converts characters into symmetric functions.
Cluster algebra. A cluster algebra is a commutative algebra generated from overlapping sets of generators called clusters, related by mutation rules. Cluster structures often appear in total positivity, Grassmannians, and canonical-basis questions.
Graded character. A graded character records the character of a graded representation degree by degree. In symmetric-function notation, this often gives a series such as \(\sum_d q^d\frobChar(M_d).\)
Hilbert series. The Hilbert series of a graded vector space records the dimensions of its homogeneous components. If the graded space carries an \(\symS_n\)-action, the Frobenius characteristic refines the Hilbert series.
Quiver. A quiver is a directed graph. Quivers enter this site mostly through quiver Hecke algebras, Hall–Littlewood-type formulas, and representation-theoretic models.
#Commuting and anticommuting variables
Bosonic variables. Bosonic variables commute: \[x_i x_j = x_j x_i.\] These are the ordinary variables used for symmetric, quasisymmetric, and polynomial bases throughout the site.
Fermionic variables. Fermionic variables anticommute: \[\theta_i\theta_j=-\theta_j\theta_i.\] In particular, over a field of characteristic zero, \(\theta_i^2=0.\) They appear in super and fermionic variants of symmetric-function and diagonal-harmonic constructions.
Plethystic notation. Plethystic notation is a compact way to substitute alphabets and virtual alphabets into symmetric functions. It is tied to the plethysm operation.
#Partitions, tableaux, and Catalan objects
Frobenius notation. The Frobenius notation of a partition records the arm and leg lengths from the diagonal boxes of its Young diagram.
Plane partition. A plane partition is a two-dimensional array of nonnegative integers with weakly decreasing rows and columns.
Tamari lattice. The Tamari lattice is a partial order on Catalan objects such as binary trees or Dyck paths. It also appears in rational, cyclic, affine, and \(\nu\)-Tamari variants; see the rational Catalan and Tamari interval discussions.
#Geometry and polytopes
Ehrhart polynomial. The Ehrhart polynomial of a lattice polytope \(P\) counts lattice points in dilates \(nP.\)
\(h^*\)-polynomial. The \(h^*\)-polynomial is the numerator of the Ehrhart series of a lattice polytope. Its coefficients are always nonnegative integers, even though the Ehrhart polynomial itself can have negative coefficients.
Hessenberg variety. A Hessenberg variety is a subvariety of a flag variety defined by a linear operator and a Hessenberg function. Regular semisimple Hessenberg varieties are connected to chromatic quasisymmetric functions through Tymoczko’s dot action.
Schubert cell. A Schubert cell is one stratum in the standard cell decomposition of a flag variety or Grassmannian. In the Grassmannian, these cells are indexed by partitions fitting inside a rectangle and are closely related to Schubert classes.