The height of a (finite) lattice L is the size of a maximum chain in L. Cameron, Solomon and Turull (1989) showed that the height of the subgroup lattice of the symmetric group S_n is given by \lceil 3n/2 \rceil - b(n), where b(n) is the number of 1s in the binary expansion of n. The analogue of S_n for semigroups is the full transformation semigroup T_n. Cameron, Gadoleau, Mitchell and Perese (2017) established an accurate asymptotic formula for the height of the subsemigroup lattice of T_n. But the subgroup lattice of a group G can be viewed from a different angle: it is (isomorphic to) the lattice of right (or left) congruences of G.(And one-sided congruences of a monoid are in a 1-1 correspondence with cyclic transformation representations of S.) In this talk I will introduce a general method which gives a lower bound for the height of the lattices of one- or two-sided congruences of an arbitrary semigroup, and under certain additional conditions gives the exact values. I will apply this theory to obtain the height for the lattices of right and left congruences of T_n, as well as for many other natural semigroups of transformations, partitions and matrices. This is joint work with Matthew Brookes, James East, Craig Miller and James Mitchell.
Heights of congruence lattices of semigroups Sponsored by the Meyer Fund
The quasisymmetric functions, QSym, are generalized for a finite alphabet A by the colored quasisymmetric functions, QSym_A, in partially commutative variables. Their dual, NSym_A, generalizes the noncommutative symmetric functions, NSym, through a relationship with Hopf algebras of trees. We define a new pair of dual bases that generalize the immaculate and dual immaculate bases of QSym and NSym. We use tableaux combinatorics and creation operators in their definitions, and we also present results on the Hopf algebra structure, skew functions, and multiplication rules. We then define a new Hopf algebra Sym_A, contained within QSym_A, that is isomorphic to the symmetric functions, Sym, when A is an alphabet of size one. Its graded dual, PSym_A, is the commutative image of NSym_A and also generalizes Sym. In addition to defining generalizations of the classic bases of the symmetric functions to Sym_A and PSym_A, we describe multiplication, comultiplication, and the antipode in terms of a basis for both algebras.
Colored Schur-like bases and generalizations of the symmetric functions
Sep. 17, 2024 3:30pm (MATH 3…
Topology
Taylor Rogers
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Chromatic homotopy theory studies the stable homotopy category (in particular, the stable homotopy groups of spheres) by means of the chromatic filtration. This filtration reveals global structure in the stable homotopy category that would be difficult to see otherwise. In this first talk, we will discuss some motivation for this filtration and introduce some tools necessary for its description and systematic study.
Heights of congruence lattices of semigroups