A.V. Borovik, Department of Mathematics, UMIST
1. Notation
This paper continues the works [1,2] and uses, with some modification, their terminology and notation. Throughout the paper W is a Coxeter group (possibly infinite) and P a finite standard parabolic subgroup of W. We identify the Coxeter group W with its Coxeter complex and refer to elements of W as chambers, to cosets with respect to a parabolic subgroup as residues, etc. We shall use the calligraphic letter as a notation for the Coxeter complex of W and the symbol for the set of left cosets of the parabolic subgroup P. We shall use the Bruhat ordering on in its geometric interpretation, as defined in [2, Theorem 5.7]. The w-Bruhat ordering on is denoted by the same symbol as the w-Bruhat ordering on . Notation , <w, >w has obvious meaning.
We refer to Tits [6] or Ronan [5] for definitions of chamber systems, galleries, geodesic galleries, residues, panels, walls, half-complexes. A short review of these concepts can be also found in [1,2].
2. Coxeter matroids
If W is a finite Coxeter group, a subset is called a Coxeter matroid (for W and P) if it satisfies the maximality property: for every the set contains a unique w-maximal element A; this means that for all . If is a Coxeter matroid we shall refer to its elements as bases. Ordinary matroids constitute a special case of Coxeter matroids, for W=Symn and P the stabiliser in W of the set [4]. The maximality property in this case is nothing else but the well-known optimal property of matroids first discovered by Gale [3].
In the case of infinite groups W we need to slightly modify the definition. In this situation the primary notion is that of a matroid map
i.e. a map satisfying the matroid inequality
The image of obviously satisfies the maximality property. Notice that, given a set with the maximality property, we can introduce the map by setting be equal to the w-maximal element of . Obviously, is a matroid map. In infinite Coxeter groups the image of the matroid map associated with a set satisfying the maximality property may happen to be a proper subset of (the set of all `extreme' or `corner' chambers of ; for example, take for a large rectangular block of chambers in the affine Coxeter group ). This never happens, however, in finite Coxeter groups, where .
So we shall call a subset a matroid if satisfies the maximality property and every element of is w-maximal in with respect to some . After that we have a natural one-to-one correspondence between matroid maps and matroid sets.
We can assign to every Coxeter matroid for W and P the Coxeter matroid for W and 1 (or W-matroid).
Теорема 1. [2, Lemma 5.15] A map
is a matroid map if and only if the map
defined by is also a matroid map.
Recall that denotes the w-maximal element in the residue . Its existence, under the assumption that the parabolic subgroup P is finite, is shown in [2, Lemma 5.14].
In is a matroid map, the map is called the underlying flag matroid map for and its image the underlying flag matroid for the Coxeter matroid . If the group W is finite then every chamber x of every residue is w-maximal in for w the opposite to x chamber of and , as a subset of the group W, is simply the union of left cosets of P belonging to .
3. Characterisation of matroid maps
Two subsets A and B in are called adjacent if there are two adjacent chambers and , the common panel of a and b being called a common panel of A and B.
Лемма 1. If A and B are two adjacent convex subsets of then all their common panels belong to the same wall .
We say in this situation that is the common wall of A and B.
For further development of our theory we need some structural results on Coxeter matroids.
Теорема 2. A map is a matroid map if and only if the following two conditions are satisfied.
(1) All the fibres , , are convex subsets of .
(2) If two fibres and of are adjacent then their images A and B are symmetric with respect to the wall containing the common panels of and , and the residues A and B lie on the opposite sides of the wall from the sets , , correspondingly.
Доказательство. If is a matroid map then the satisfaction of conditions (1) and (2) is the main result of [2].
Assume now that satisfies the conditions (1) and (2).
First we introduce, for any two adjacent fibbers and of the map , the wall separating them. Let be the set of all walls .
Now take two arbitrary residues and chambers and . We wish to prove .
Consider a geodesic gallery
connecting the chambers u and v. Let now the chamber x moves along from u to v, then the corresponding residue moves from to . Since the geodesic gallery intersects every wall no more than once [5, Lemma 2.5], the chamber x crosses each wall in no more than once and, if it crosses , it moves from the same side of as u to the opposite side. But, by the assumptions of the theorem, this means that the residue crosses each wall no more than once and moves from the side of opposite u to the side containing u. But, by the geometric interpretation of the Bruhat order, this means [2, Theorem 5.7] that decreases, with respect to the u-Bruhat order, at every such step, and we ultimately obtain
Список
литературы
Borovik A.V., Gelfand I.M. WP-matroids and thin Schubert cells on Tits systems // Advances Math. 1994. V.103. N.1. P.162-179.
Borovik A.V., Roberts K.S. Coxeter groups and matroids, in Groups of Lie Type and Geometries, W. M. Kantor and L. Di Martino, eds. Cambridge University Press. Cambridge, 1995 (London Math. Soc. Lect. Notes Ser. V.207) P.13-34.
Gale D., Optimal assignments in an ordered set: an application of matroid theory // J. Combinatorial Theory. 1968. V.4. P.1073-1082.
Gelfand I.M., Serganova V.V. Combinatorial geometries and torus strata on homogeneous compact manifolds // Russian Math. Surveys. 1987. V.42. P.133-168.
Ronan M. Lectures on Buildings - Academic Press. Boston. 1989.
Tits J. A local approach to buildings, in The Geometric Vein (Coxeter Festschrift) Springer-Verlag, New York a.o., 1981. P.317-322.
Другие работы по теме:
Лаури-Пуэбло
Лаури-Пуэбло , англ. Lowry Pueblo — археологический памятник древних пуэбло на территории Национального памятника «Каньоны древних» (en:Canyons of the Ancients National Monument) близ Плезант-Вью в штате Колорадо, США. Поселение-пуэбло было построено около 1060 г. поверх колодцеобразных домов более древнего происхождения.
Планисфера Кантино
— драгоценный памятник эпохи Великих географических открытий, запечатлевший состояние географических знаний португальцев на рубеже XV и XVI веков.
The Mambo Kings Play Songs Of Love
Essay, Research Paper The Mambo Kings Play Songs of Love By: Oscar Hijuelo Harper Collins Publishers 1989 407 Pages (No Bibliography, Pics, Maps or Index)
Syllabus
–American Poetry From 1900-1950 Essay, Research Paper Cary Nelson Except as noted, all poems are in Anthology of Modern American Poetry (Oxford). All authors have web sites on MAPS. Before each
Early Day Astronomers And Their Influences In
Astronomy Essay, Research Paper In modern day astronomy and the study of the universe, individuals usually come up with predictions and then gather information and data in order to prove or disprove their hypothesis. Other astronomers in the history of astronomy did the same and came up with their own conclusions based on the data they collected.
The Inflationary Universe Essay Research Paper 1What
The Inflationary Universe Essay, Research Paper 1.What is the most interesting thing you have learned from this book? Alan Guth s book takes you through the journey of his life, and his career in which he made many scientific discoveries. I especially liked the way that he maps out these discoveries because it was fun to discover along with him.
Counter Strike Essay Research Paper There have
Counter Strike Essay, Research Paper There have been many significant new additions to the entity set for Beta5. The following is a list of new features and how to implement them. All Beta4 entities are still supported and their functionality has not been changed. If you add the new Beta5 entities they will be used instead of the Beta4 entities you had in the map before.
Government Allowing Marijuana Use For Aids Essay
, Research Paper Marijuana-for-AIDS Study? The first domestic study of marijuana’s effects on AIDS patients is expected to begin in May in San Francisco. While this preliminary study is designed to focus on whether marijuana is safe enough for AIDS patients to use, it is the first clinical trial in 15 years that would obtain any data whatsoever on marijuana’s medical effectiveness.
Internet A World Of Its Own Essay
, Research Paper The Internet, a World of its Own Thinking about the internet reminds me of something that was said in my history class. My professor was lecturing about the early colonies in America and how the men of that time had to sail up and down coasts in order to make maps. They had to actually draw these landscapes as they saw them.
The Qualities Of A Prince Essay Research
Paper The Qualities of the Prince is Machiavelli’s “how to” manual for a great leader. He maps out each quality so that it may be easily be implemented by the aspiring emperor, king, or dictator. His premise is that there are some things that must be sacrificed, such as high morals, in order to gain others, such as power.
Teaching Content Area Vocabulary Essay Research Paper
To be successful readers of both narrative and expository text, students must have intense vocabulary instruction. If children have a wide range of vocabulary knowledge, then they can better interpret the text they read. Because experience helps form students vocabulary base, teachers must provide these experiences, both directly and vicariously, to their students.
Lyon Hypothesis And CrossOver Essay Research Paper
These two ideas are similar and different from one another for a variety of things. The Lyon hypothesis, also called the X inactivation hypothesis, maintains that in the early embryonic development of a normal female, one X chromosome becomes inactivated in each body cell. Chance is used to show which of the two X chromosomes is inactivated in a given cell.
Great Plains Essay Research Paper For this
Great Plains Essay, Research Paper For this assignment we are asked mainly to answer the questions given to use over the reading material, as a measure of accounting for each student completing the reading. In class we have briefly discussed these articles to the extent to figure out that most saw the article written by Walter Webb as being less than factual from a Historical context.
Sense And Reference Essay Research Paper April
Sense And Reference Essay, Research Paper April Chlumsky Philosophy 3/4/00 Frege on Sense and Reference Gottlob Frege developed the theory of sense and reference into a thorough philosophy of language. He assigned logic the task of discovering the laws of truth, not of assertion or thought. The work ?On Sense and Reference? has received a great deal of attention throughout the years.
Francis Drake Essay Research Paper Author John
Francis Drake Essay, Research Paper Author John Guy Drake, a 16th Century Explorer Ticktock Publishing-1997 SIR FRANCIS DRAKE Sir Francis Drake is known for England’s defeat of the Spanish Armada. He was a sea dog and he sailed around the world. Drake was actually the first one to sail around the world since Magellan died in his journey.
Water Quality
– Lab#4 Essay, Research Paper Lab #4 Oakland Ravine and Lake: Water Quality 1. The pH levels are fairly constant for inlet and outlet. The dissolved oxygen(dO), on the other hand, varies very much between the inlet and the outlet. The measurements of the pH and the dO of the water do not differ greatly. By comparing the two graphs given I can see that both the pH and the dissolved oxygen are lover in the winter.
The Four Color Theorem Essay Research
Paper The Four Color Theorem Throughout history, mapmakers have perfected the art of making maps that allow for viewers to differentiate between distinct regions, such as countries or states. With any mapmakers given knowledge and practice, it has become known that if you plan well enough, you will never need more than four colors to color the maps that you make.
Italy Essay Research Paper The Internet Tourist
Italy Essay, Research Paper The Internet Tourist Guide Of Italy – A city by city guide to Italy. Also has maps and transportation information. TravelEurope’s Italy Travel Guide – More than 4000 web pages about Italy including art,
Legalization Of Medicinal Marijuana Essay Research Paper
Legalization of Marijuana During the Carter, Reagan, and Bush administrations, eight people in the United States were allowed to use marijuana for medicinal purposes under the Compassionate Investigative New Drug program. However, since the Clinton administration no new applications have been accepted.
Untitled Essay Research Paper Ancient History The
Untitled Essay, Research Paper Ancient History The evolution of the world The Greek geographers of the later Roman period developed systematic calculations for the mapping and shaping of the earth. However, what would come to replace these systematic calculations? Why were the ideologies of a flat earth accepted and why were those of a spherical earth ridiculed? The answer to this question is very simple and can be answered by one clear and concise word: Religion. “Thus saith the Lord God; This is Jerusalem: I have set it in the midst of the nations and countries that are round about her.” (Ezekiel 5:5) This verse from the of book Ezekiel simply states that the city of Jerusalem should be in the center of all maps created.
Human Genome Project Essay Research Paper What
Human Genome Project Essay, Research Paper What is the Human Genome Project? The Human Genome Project (HGP) is an international research program designed to construct detailed genetic and physical maps of the
Discussion Of Waiting For Godo Essay Research
Paper Discussion Paper #7 Waiting for Godot Samuel Beckett s Waiting for Godot is a play without meaning. Vladimir (Didi) and Estragon (Gogo) go on for pages with meaningless jibber-jabber. The setting is an obscure place with no distinguishable characteristics; there is only a tree and a road to decorate the mundane landscape.
Nature V Nurture 2 Essay Research Paper
The nature vs. nurture debate, long contested by philosophers and scientists alike, is of special interest tothose charged with the responsibility of raising kids in a complicated, ever-changing world. While today’s mothers and fathers must navigate the same child-rearing seas as their own parents did, they must alsocontend with the wide range of conflicting theory and advice recently generated by this volatile issue.
Hiking To Understanding Essay Research Paper Hiking
Hiking To Understanding Essay, Research Paper Hiking to Understanding Surrounded by thousands of stars, complete silence, and spectacular mountains, I stood atop New Hampshire’s Presidential Range awestruck by nature’s beauty. Immediately, I realized that I must dedicate my life to understanding the causes of the universe’s beauty.
The Human Genome Project Essay Research Paper
The Human Genome Project The Human Genome Project (HGP) is an international research effort to chart and characterize the human genome; the entire package of genetic instructions for a human being. That entails laying out, in order the 3 billion DNA letters (or base pairs) of the full human genetic code.
Gps Essay Research Paper When cars were
Gps Essay, Research Paper When cars were first invented, something like an electronic map that “pin-points” you on the globe and could find the shortest route for you to take to get to your destination in your car would be rediculous, and would only be dreamed about
Geographic Information Systems Essay Research Paper Geographic
Geographic Information Systems Essay, Research Paper Geographic Information Systems Geographic information systems (GIS) technology can be used for scientific investigations, resource management, and development planning. For example, a GIS might allow emergency planners to easily calculate emergency response times and effected areas of the ocean during an oil spill based on the spills location.
Gene Mapping Essay Research Paper Gene Mapping
Gene Mapping Essay, Research Paper Gene Mapping began when the U.S. Government held a conference to explore if DNA damage occurred in people exposed to low levels of radiation in Japan after the 1945 Atomic Bombs. There, scientists quickly realized that observing the human genome could be useful in discovering environmental mutates.
Computer technology
Development of computer technologies. Machines, which are able to be learned from experience and not forget that they studied, and able to work unassisted or control of man. Internet as global collection of different types of computer networks.