# Finite Coxeter Groups and Generalized Elnitsky Tilings

@inproceedings{Nicolaides2021FiniteCG, title={Finite Coxeter Groups and Generalized Elnitsky Tilings}, author={Robert Nicolaides and Peter Rowley}, year={2021} }

In [5], Elnitsky constructed three elegant bijections between classes of reduced words for Type A, B and D families of Coxeter groups and certain tilings of polygons. This paper oﬀers a particular generalization of this concept to all ﬁnite Coxeter Groups in terms of embeddings into the Symmetric Group.

## References

SHOWING 1-10 OF 12 REFERENCES

### Subtilings of Elnitsky Tilings for Finite Irreducible Coxeter Groups

- Mathematics
- 2021

Two new Elnitsky tilings for Coxeter groups of type B are displayed as certain subtilings. Additionally, a new tiling for the non-crystallographic Coxeter group of type H3 is obtained, described as a…

### Combinatorics of Coxeter Groups

- Mathematics
- 2005

I.- The basics.- Bruhat order.- Weak order and reduced words.- Roots, games, and automata.- II.- Kazhdan-Lusztig and R-polynomials.- Kazhdan-Lusztig representations.- Enumeration.- Combinatorial…

### Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups

- MathematicsJ. Comb. Theory, Ser. A
- 1997

A bijection is established between the resulting equivalence classes and rhombic tilings of a certain 2n-gon determined by the permutation of the Coxeter groups, namely those of typesBandD.

### On the Number of Reduced Decompositions of Elements of Coxeter Groups

- MathematicsEur. J. Comb.
- 1984

### Rhombic tilings and Bott-Samelson varieties

- Mathematics
- 2016

S.~Elnitsky (1997) gave an elegant bijection between rhombic tilings of $2n$-gons and commutation classes of reduced words in the symmetric group on $n$ letters. P.~Magyar (1998) found an important…

### Fully commutative elements in finite and affine Coxeter groups

- Mathematics
- 2014

An element of a Coxeter group $$W$$W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were…

### On the Fully Commutative Elements of Coxeter Groups

- Mathematics
- 1996

Let W be a Coxeter group. We define an element w ∈ W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting…

### Rhombus Tilings of an Even-Sided Polygon and Quadrangulations on the Projective Plane

- MathematicsGraphs Comb.
- 2020

It is proved that the rhombus tilings of a regular 2 k -gon bijectively correspond to the pairs of a k -minimal quadrangulations on the projective plane and its specified noncontractible k -cycle.

### On the number of commutation classes of the longest element in the symmetric group

- Mathematics
- 2016

Using the standard Coxeter presentation for the symmetric group $S_n$, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from…