In mathematics, Young's lattice is a lattice that is formed by all integer partitions. 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 diagrams and the partial order 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 algebras.
Young's lattice is a lattice (and hence also a partially ordered set) Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers diagrams).
The traditional application of Young's lattice is to the description of the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. The equivalence classes of irreducible representations may be parametrized by partitions or Young diagrams, the restriction from Sn +1 to Sn is multiplicity-free, and the representation of Sn with partition p is contained in the representation of Sn +1 with partition q if and only if q covers p in Young's lattice. Iterating this procedure, one arrives at Young's semicanonical basis in the irreducible representation of Sn with partition p, which is indexed by the standard Young tableaux of shape p.
The poset 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 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.
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Motivated by the recent generalization of the Haldane conjecture to SU(3) chains [Lajko et al., Nucl. Phys. B924, 508 (2017)] according to which a Haldane gap should be present for symmetric representations if the number of boxes in the Young diagram is a ...
In this thesis we study various one-dimensional quantum spin systems with SU(2) and SU(N) symmetry. We investigate the short-distance behavior of the SU(2) Heisenberg model in the limit of large spin and show that there exists an extended regime where pert ...
EPFL2020
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It was recently argued that SU(3) chains in the p-box symmetric irreducible representation (irrep) exhibit a "Haldane gap" when p is a multiple of 3 and are otherwise gapless [Nucl. Phys. B 924, 508 (2017)]. We extend this argument to the self-conjugate ir ...
In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram. Let be a set with a partial order . As usual, let be the relation on such that if and only if and . Let and be elements of . Then covers , written , if and there is no element such that .
In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1 The only partition of zero is the empty sum, having no parts.
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra.