In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Young's lattice is a
lattice that is formed by all
integer partitions
In number theory and combinatorics, a partition of a non-negative integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same ...
. It is named after
Alfred Young, who, in a series of papers ''On quantitative substitutional analysis,'' developed the
representation theory of the symmetric group. In Young's theory, the objects now called
Young diagram
In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups a ...
s and the
partial order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
on them played a key, even decisive, role. Young's lattice prominently figures in
algebraic combinatorics, forming the simplest example of a
differential poset in the sense of . It is also closely connected with the
crystal bases for
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody ...
s.
Definition
Young's lattice is a
lattice (and hence also a
partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
) ''Y'' formed by all integer partitions ordered by inclusion of their Young diagrams (or
Ferrers diagrams).
Significance
The traditional application of Young's lattice is to the description of the
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s of
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
s S
''n'' for all ''n'', together with their branching properties, in
characteristic zero. The
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of irreducible representations may be parametrized by partitions or Young diagrams, the restriction from S
''n''+1 to S
''n'' is multiplicity-free, and the representation of S
''n'' with partition ''p'' is contained in the representation of S
''n''+1 with partition ''q''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
''q'' covers ''p'' in Young's lattice. Iterating this procedure, one arrives at ''Young's semicanonical basis'' in the irreducible representation of S
''n'' with partition ''p'', which is indexed by the standard Young tableaux of shape ''p''.
Properties
* The
poset
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
''Y'' is
graded: the minimal element is ∅, the unique partition of zero, and the partitions of ''n'' have rank ''n''. This means that given two partitions that are comparable in the lattice, their ranks are ordered in the same sense as the partitions, and there is at least one intermediate partition of each intermediate rank.
* The poset ''Y'' is a lattice. The
meet and join
In mathematics, specifically order theory, the join of a subset S of a partially ordered set P is the supremum (least upper bound) of S, denoted \bigvee S, and similarly, the meet of S is the infimum (greatest lower bound), denoted \bigwedge S. ...
of two partitions are given by the intersection and the union of the corresponding Young diagrams. Because it is a lattice in which the meet and join operations are represented by intersections and unions, it is a
distributive lattice
In mathematics, a distributive lattice is a lattice (order), lattice in which the operations of join and meet distributivity, distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice o ...
.
* If a partition ''p''
covers ''k'' elements of Young's lattice for some ''k'' then it is covered by ''k'' + 1 elements. All partitions covered by ''p'' can be found by removing one of the "corners" of its Young diagram (boxes at the end both of their row and of their column). All partitions covering ''p'' can be found by adding one of the "dual corners" to its Young diagram (boxes outside the diagram that are the first such box both in their row and in their column). There is always a dual corner in the first row, and for each other dual corner there is a corner in the previous row, whence the stated property.
* If distinct partitions ''p'' and ''q'' both cover ''k'' elements of ''Y'' then ''k'' is 0 or 1, and ''p'' and ''q'' are covered by ''k'' elements. In plain language: two partitions can have at most one (third) partition covered by both (their respective diagrams then each have one box not belonging to the other), in which case there is also one (fourth) partition covering them both (whose diagram is the union of their diagrams).
* Saturated chains between ∅ and ''p'' are in a natural
bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
with the standard
Young tableaux
In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups ...
of shape ''p'': the diagrams in the chain add the boxes of the diagram of the standard Young tableau in the order of their numbering. More generally, saturated chains between ''q'' and ''p'' are in a natural bijection with the skew standard tableaux of
skew shape ''p''/''q''.
* The
Möbius function
The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and m ...
of Young's lattice takes values 0, ±1. It is given by the formula
::
Dihedral symmetry
Conventionally, Young's lattice is depicted in a
Hasse diagram
In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set (S,\le) one represents each ...
with all elements of the same rank shown at the same height above the bottom.
has shown that a different way of depicting some subsets of Young's lattice shows some unexpected symmetries.
The partition
:
of the ''n''th
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
has a
Ferrers diagram that looks like a staircase. The largest elements whose Ferrers diagrams are rectangular that lie under the staircase are these:
:
Partitions of this form are the only ones that have only one element immediately below them in Young's lattice. Suter showed that the set of all elements less than or equal to these particular partitions has not only the bilateral symmetry that one expects of Young's lattice, but also rotational symmetry: the rotation group of order ''n'' + 1
acts on this poset. Since this set has both bilateral symmetry and rotational symmetry, it must have dihedral symmetry: the (''n'' + 1)st
dihedral group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
acts
faithfully on this set. The size of this set is 2
''n''.
For example, when ''n'' = 4, then the maximal element under the "staircase" that have rectangular Ferrers diagrams are
: 1 + 1 + 1 + 1
: 2 + 2 + 2
: 3 + 3
: 4
The subset of Young's lattice lying below these partitions has both bilateral symmetry and 5-fold rotational symmetry. Hence the dihedral group ''D''
5 acts faithfully on this subset of Young's lattice.
See also
*
Young–Fibonacci lattice
*
Bratteli diagram
References
*
*
*
*
{{Order theory
Integer partitions
Lattice theory
Representation theory
Symmetric functions