Young Lattice Research Articles

Complexity in Young's lattice

We investigate the complexity of the partial order relation of Young's lattice. The definable relations are characterized by establishing the maximal definability property modulo the single automorphism given by conjugation; consequently, as an ordered set Young's lattice has an undecidable elementary theory and is non-finitely axiomatizable but every ideal generates a finitely axiomatizable universal class of equivalence relations. We show the class of equivalence relations has positive definability and is second-order capable in the finite. We end with conjectures concerning the complexities of the Σ1 and Σ2-theories of Young's lattice.

Up- and Down-Operators on Young's Lattice

The up-operators $u_i$ and down-operators $d_i$ (introduced as Schur operators by Fomin) act on partitions by adding/removing a box to/from the $i$th column if possible. It is well known that the $u_i$ alone satisfy the relations of the (local) plactic monoid, and the present authors recently showed that relations of degree at most 4 suffice to describe all relations between the up-operators. Here we characterize the algebra generated by the up- and down-operators together, showing that it can be presented using only quadratic relations.

An Order on Circular Permutations

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

Universal adiabatic quantum computation via the space-time circuit-to-Hamiltonian construction.

We show how to perform universal adiabatic quantum computation using a Hamiltonian which describes a set of particles with local interactions on a two-dimensional grid. A single parameter in the Hamiltonian is adiabatically changed as a function of time to simulate the quantum circuit. We bound the eigenvalue gap above the unique ground state by mapping our model onto the ferromagnetic XXZ chain with kink boundary conditions; the gap of this spin chain was computed exactly by Koma and Nachtergaele using its q-deformed version of SU(2) symmetry. We also discuss a related time-independent Hamiltonian which was shown by Janzing to be capable of universal computation. We observe that in the limit of large system size, the time evolution is equivalent to the exactly solvable quantum walk on Young's lattice.

Symmetries of the k-bounded partition lattice

We generalize the symmetry on Young's lattice, found by Suter, to a symmetry on the $k$-bounded partition lattice of Lapointe, Lascoux and Morse. Nous généralisons la symétrie sur le treillis de Young, découvert par Suter, à une symétrie sur le treillis des partages bornés par $k$ et étudié par Lapointe, Lascoux and Morse.

Parabolic Kazhdan–Lusztig R-polynomials for Hermitian symmetric pairs

We give explicit combinatorial product formulas for the parabolic Kazhdan–Lusztig R-polynomials of Hermitian symmetric pairs. Our results imply that all the roots of these polynomials are (either zero or) roots of unity, and complete those in [F. Brenti, Kazhdan–Lusztig and R-polynomials, Young's lattice, and Dyck partitions, Pacific J. Math. 207 (2002) 257–286] on Hermitian symmetric pairs of type A. As an application of our results, we derive explicit combinatorial product formulas for certain sums and alternating sums of ordinary Kazhdan–Lusztig R-polynomials.

Signed differential posets and sign-imbalance

We study signed differential posets, a signed version of differential posets. These posets satisfy enumerative identities which are signed analogues of those satisfied by differential posets. Our main motivations are the sign-imbalance identities for partition shapes originally conjectured by Stanley, now proven in [T. Lam, Growth diagrams, domino insertion and sign-imbalance, J. Combin. Theory Ser. A 107 (2004) 87–115; A. Reifergerste, Permutation sign under the Robinson–Schensted–Knuth correspondence, Ann. Comb. 8 (2004) 103–112; J. Sjöstrand, On the sign-imbalance of partition shapes, J. Combin. Theory Ser. A 111 (2005) 190–203]. We show that these identities result from a signed differential poset structure on Young's lattice, and explain similar identities for Fibonacci shapes.

Counting paths in Young's lattice

Young's lattice is the lattice of partitions of integers, ordered by inclusion of diagrams. Standard Young tableaux can be represented as paths in Young's lattice that go up by one square at each step, and more general paths in Young's lattice correspond to more general kinds of tableaux. Using the theory of symmetric functions, in particular Pieri's rule for multiplying a Schur function by a complete symmetric function, we derive formulas for counting paths in Young's lattice that go up or down by horizontal or vertical strips. Our results are related to Richard Stanley's theory of differential posets in the special case of Young's lattice.

Unimodality and Young's lattice

Young's lattice of a partition λ consists of all partitions whose Ferrers diagrams fit inside λ. Several infinite families of partitions are given whose Young's lattice is not rank unimodal. Some related problems are discussed.

Greedy matching in Young's lattice

If the level sets of a ranked partially ordered set are totally ordered, the greedy match between adjacent levels is defined by successively matching each vertex on one level to the first available unmatched vertex, if any, on the next level. Aigner showed that the greedy match produces symmetric chains in the Boolean algebra. We extend that result to partially ordered sets which are products of chains.