Steinberg representation

Steinberg representation

In mathematics, the Steinberg representation, or Steinberg module, denoted by "St", is a particular linear representation of a group of Lie type over a finite field of characteristic "p", of degree equal to the largest power of "p" dividing the order of the group. These representations were discovered by harvs|txt=yes|authorlink=Robert Steinberg|first=Robert |last=Steinberg|year=1957.

Most finite simple groups have exactly one Steinberg representation. A few have more than one because they are groups of Lie type in more than one way, and sporadic and most alternating groups have no Steinberg representation.

Properties

*The character value of "St" on an element "g" equals, up to sign, the order of a Sylow subgroup of the centralizer of "g" if "g" has order prime to "p", and is zero if the order of "g" is divisible by "p".
*The Steinberg representation is equal to an alternating sum over all parabolic subgroups containing a Borel subgroup, of the representation induced from the identity representation of the parabolic subgroup.
*The Steinberg representation is both regular and unipotent, and is the only irreducible regular unipotent representation (for the given prime "p").

Applications

*The Steinberg representation is used in the proof of Haboush's theorem (the Mumford conjecture).

References

* "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters" (Wiley Classics Library) by Roger W. Carter, John Wiley & Sons Inc; New Ed edition (August 1993) ISBN 0-471-94109-3
*springer|id=S/s130530|title= Steinberg module|first=Robert |last=Steinberg
*citation|first=R. |last=Steinberg|title=Prime power representations of finite linear groups II |journal=Canad. J. Math. |volume= 9 |year=1957|pages=347-351
*R. Steinberg, "Collected Papers" , Amer. Math. Soc. (1997) ISBN 0-8218-0576-2 pp. 580–586
*citation|first=J.E.|last= Humphreys|title=The Steinberg representation|journal= Bull. Amer. Math. Soc. (N.S.) |volume= 16 |year=1987|pages=237–263
url=http://www.ams.org/bull/1987-16-02/S0273-0979-1987-15512-1/home.html
id=MR|876960


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Steinberg (disambiguation) — Steinberg is a German musical equipment and software company.The name Steinberg (German for mountain of stone ) may also refer to:PeopleMusic*Billy Steinberg, American songwriter *Elliot Easton (born Elliot Steinberg, 1953), American musician… …   Wikipedia

  • Steinberg v. Columbia Pictures Industries, Inc. — Steinberg v. Columbia Pictures Industries, Inc., 663 F. Supp. 706 (S.D.N.Y. 1987) was a federal case in which artist Saul Steinberg sued various parties involved with producing and promoting the 1984 movie Moscow on the Hudson , claiming that a… …   Wikipedia

  • STEINBERG (S.) — STEINBERG SAUL (1914 ) Si l’on reconnaît la plupart des artistes à leur style, Steinberg, lui, n’est identifiable qu’à travers la diversité de ses styles, qu’à travers son infidélité permanente à l’égard de ses propres productions. Il ne semble… …   Encyclopédie Universelle

  • Steinberg Cubase — Cubase Cubase 6 Developer(s) Steinberg …   Wikipedia

  • Robert Steinberg — (born May 251922 Soroki, Romania) is a mathematician at the University of California, Los Angeles who invented the Steinberg representation, the Steinberg group in algebraic K theory, and the Steinberg groups in Lie theory that yield finite… …   Wikipedia

  • Unipotent representation — In mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups. Informally, Langlands philosophy suggests that there should be a correspondence between… …   Wikipedia

  • Leigh Steinberg Sports & Entertainment — (LSSE) is a full service representation and marketing firm specializing in the careers of professional athletes, olympians, coaches, and television/radio personalities. The firm is run by agent Leigh Steinberg who is widely considered a pioneer… …   Wikipedia

  • Deligne–Lusztig theory — In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ adic cohomology with compact support, introduced by Deligne Lusztig (1976). Lusztig (1984) used these representations to… …   Wikipedia

  • Haboush's theorem — In mathematics Haboush s theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K , and for any linear representation ρ of G on a K vector space V , given v ne;0 in V that is… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”