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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 ...
, a Coxeter element is an element of an irreducible
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same
order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...
. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups.


Definitions

Note that this article assumes a finite
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
. For infinite Coxeter groups, there are multiple
conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
es of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number of an irreducible root system. *The Coxeter number is the order of any Coxeter element;. *The Coxeter number is where is the rank, and is the number of reflections. In the crystallographic case, is half the number of
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...
; and is the dimension of the corresponding semisimple
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
. *If the highest root is \sum m_i \alpha_i for simple roots , then the Coxeter number is 1 + \sum m_i. *The Coxeter number is the highest degree of a fundamental invariant of the Coxeter group acting on polynomials. The Coxeter number for each Dynkin type is given in the following table: The invariants of the Coxeter group acting on polynomials form a polynomial algebra whose generators are the fundamental invariants; their degrees are given in the table above. Notice that if is a degree of a fundamental invariant then so is . The eigenvalues of a Coxeter element are the numbers e^ as runs through the degrees of the fundamental invariants. Since this starts with , these include the primitive th root of unity, \zeta_h = e^, which is important in the Coxeter plane, below. The dual Coxeter number is 1 plus the sum of the coefficients of simple roots in the highest short root of the dual root system.


Group order

There are relations between the order of the Coxeter group and the Coxeter number : \begin & \quad \frac = 1 \\ pt ,q& \quad \frac = \frac + \frac -1 \\ pt ,q,r& \quad \frac = 12 - p - 2q - r + \frac + \frac \\ pt ,q,r,s& \quad \frac = \frac + \frac + \frac - \frac - \frac - \frac - \frac +1 \\ pt \vdots \qquad & \qquad \vdots \end For example, has : \begin &\frac = 12 - 3 - 6 - 5 + \frac + \frac = \frac, \\ pt &\therefore g_ = \frac = 960 \times 15 = 14400. \end


Coxeter elements

Distinct Coxeter elements correspond to orientations of the Coxeter diagram (i.e. to Dynkin quivers): the simple reflections corresponding to source vertices are written first, downstream vertices later, and sinks last. (The choice of order among non-adjacent vertices is irrelevant, since they correspond to commuting reflections.) A special choice is the alternating orientation, in which the simple reflections are partitioned into two sets of non-adjacent vertices, and all edges are oriented from the first to the second set. The alternating orientation produces a special Coxeter element satisfying w^= w_0, where is the longest element, provided the Coxeter number is even. For A_ \cong S_n, the
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 ...
on elements, Coxeter elements are certain -cycles: the product of simple reflections (1,2) (2,3) \cdots (n-1,n) is the Coxeter element (1,2,3,\dots, n). For even, the alternating orientation Coxeter element is: (1,2)(3,4)\cdots (2,3)(4,5) \cdots = (2,4,6,\ldots,n2,n, n1,n3,\ldots,5,3,1). There are 2^ distinct Coxeter elements among the (n1)! -cycles. The
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 ...
is generated by two reflections that form an angle of \tfrac, and thus the two Coxeter elements are their product in either order, which is a rotation by \pm \tfrac.


Coxeter plane

For a given Coxeter element , there is a unique plane on which acts by rotation by This is called the Coxeter plane and is the plane on which has eigenvalues e^ and e^ = e^. This plane was first systematically studied in , and subsequently used in to provide uniform proofs about properties of Coxeter elements. The Coxeter plane is often used to draw diagrams of higher-dimensional polytopes and root systems – the vertices and edges of the polytope, or roots (and some edges connecting these) are orthogonally projected onto the Coxeter plane, yielding a
Petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...
with -fold rotational symmetry. For root systems, no root maps to zero, corresponding to the Coxeter element not fixing any root or rather axis (not having eigenvalue 1 or −1), so the projections of orbits under form -fold circular arrangements and there is an empty center, as in the diagram at above right. For polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s. In three dimensions, the symmetry of a
regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitive group action, transitively on its Flag (geometry), flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In ...
, with one directed Petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry , , order . Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, , , order . In orthogonal 2D projection, this becomes
dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...
, , , order . In four dimensions, the symmetry of a regular polychoron, with one directed Petrie polygon marked is a
double rotation In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article ''rotation'' means ''rotational dis ...
, defined as a composite of 4 reflections, with symmetry ( John H. Conway), (#1',
Patrick du Val Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named af ...
(1964)Patrick Du Val, ''Homographies, quaternions and rotations'', Oxford Mathematical Monographs,
Clarendon Press Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books ...
,
Oxford Oxford () is a City status in the United Kingdom, cathedral city and non-metropolitan district in Oxfordshire, England, of which it is the county town. The city is home to the University of Oxford, the List of oldest universities in continuou ...
, 1964.
), order . In five dimensions, the symmetry of a regular 5-polytope, with one directed Petrie polygon marked, is represented by the composite of 5 reflections. In dimensions 6 to 8 there are 3 exceptional Coxeter groups; one uniform polytope from each dimension represents the roots of the exceptional Lie groups . The Coxeter elements are 12, 18 and 30 respectively. {, class=wikitable , + {{math, E''n'' groups , - !Coxeter group !{{math, E{{sub, 6 !{{math, E{{sub, 7 !{{math, E{{sub, 8 , - align=center !Graph ,
122
{{CDD, nodea, 3a, nodea, 3a, branch_01lr, 3a, nodea, 3a, nodea ,
231
{{CDD, nodea, 3a, nodea, 3a, nodea, 3a, branch, 3a, nodea, 3a, nodea_1 ,
421
{{CDD, nodea_1, 3a, nodea, 3a, nodea, 3a, nodea, 3a, branch, 3a, nodea, 3a, nodea , - align=center !Coxeter plane
symmetry , {{math, Dih12, 2 (*12•) , {{math, Dih18, 8 (*18•) , {{math, Dih30, 0 (*30•)


See also

*
Longest element of a Coxeter group In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by ''w''0. Properties * A C ...


Notes

{{reflist


References

{{refbegin *{{Citation , last = Coxeter , first = H. S. M. , author-link = H. S. M. Coxeter , title =
Regular Polytopes ''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...
, publisher = Methuen and Co. , year = 1948 *{{Citation , doi = 10.1090/S0002-9947-1959-0106428-2 , issn = 0002-9947 , volume = 91 , issue = 3 , pages = 493–504 , last = Steinberg , first = R. , title = Finite Reflection Groups , journal =
Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...
, date=June 1959 , jstor = 1993261 , doi-access = free *Hiller, Howard ''Geometry of Coxeter groups.'' Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. {{ISBN, 0-273-08517-4 * {{citation , first=James E. , last=Humphreys , title=Reflection Groups and Coxeter Groups , pages=74–76 (Section 3.16, ''Coxeter Elements'') , publisher=
Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, year=1992 , isbn=978-0-521-43613-7 , url=https://books.google.com/books?id=ODfjmOeNLMUC *{{citation , title = Coxeter Planes , url = http://www.math.lsa.umich.edu/~jrs/coxplane.html , date = April 9, 2007 , first = John , last = Stembridge , access-date = April 21, 2010 , archive-url = https://web.archive.org/web/20180210123511/http://www.math.lsa.umich.edu/~jrs/coxplane.html , archive-date = February 10, 2018 , url-status = dead * {{Citation , title = Notes on Coxeter Transformations and the McKay Correspondence , series = Springer Monographs in Mathematics , first = R. , last = Stekolshchik , year = 2008 , isbn = 978-3-540-77398-6 , doi = 10.1007/978-3-540-77399-3 , arxiv = math/0510216 , s2cid = 117958873 *{{citation , journal = Séminaire Lotharingien de Combinatoire , volume = B63b , year = 2010 , pages = 32 , first = Nathan , last = Reading , title = Noncrossing Partitions, Clusters and the Coxeter Plane , url = http://www.emis.de/journals/SLC/wpapers/s63reading.html *Bernšteĭn, I. N.; Gelʹfand, I. M.; Ponomarev, V. A., "Coxeter functors, and Gabriel's theorem" (Russian), ''Uspekhi Mat. Nauk'' 28 (1973), no. 2(170), 19–33
Translation on Bernstein's website
{{refend Lie groups Coxeter groups