View source for Relative root system

''of a connected reductive algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810301.png" /> defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810302.png" />''

A system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810303.png" /> of non-zero weights of the adjoint representation of a maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810304.png" />-split torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810305.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810306.png" /> in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810307.png" /> of this group (cf. [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]). The weights themselves are called the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810308.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r0810309.png" />. The relative root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103010.png" />, which can be seen as a subset of its linear envelope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103011.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103013.png" /> is the group of rational characters of the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103014.png" />, is a [[Root system|root system]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103015.png" /> be the normalizer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103016.png" /> the centralizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103018.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103019.png" /> is the connected component of the unit of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103020.png" />; the finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103021.png" /> is called the Weyl group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103022.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103023.png" />, or the relative Weyl group. The adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103025.png" /> defines a linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103027.png" />. This representation is faithful and its image is the [[Weyl group|Weyl group]] of the root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103028.png" />, which enables one to identify these two groups. Since two maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103029.png" />-split tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103032.png" /> are conjugate over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103033.png" />, the relative root systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103034.png" /> and the relative Weyl groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103036.png" />, are isomorphic, respectively. Hence they are often denoted simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103038.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103039.png" /> is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103040.png" />, the relative root system and the relative Weyl group coincide, respectively, with the usual (absolute) root system and Weyl group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103041.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103042.png" /> be the weight subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103043.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103044.png" />, corresponding to the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103045.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103046.png" /> is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103047.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103048.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103049.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103050.png" /> is a reduced root system; this is not so in general: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103051.png" /> does not have to be reduced and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103052.png" /> can be greater than 1. The relative root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103053.png" /> is irreducible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103054.png" /> is simple over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103055.png" />.

The relative root system plays an important role in the description of the structure and in the classification of semi-simple algebraic groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103056.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103057.png" /> be semi-simple, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103058.png" /> be a maximal torus defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103059.png" /> and containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103060.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103062.png" /> be the groups of rational characters of the tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103064.png" /> with fixed compatible order relations, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103065.png" /> be a corresponding system of simple roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103066.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103067.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103068.png" /> be the subsystem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103069.png" /> consisting of the characters which are trivial on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103070.png" />. Moreover, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103071.png" /> be the system of simple roots in the relative root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103072.png" /> defined by the order relation chosen on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103073.png" />; it consists of the restrictions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103074.png" /> of the characters of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103075.png" />. The Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103076.png" /> acts naturally on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103077.png" />, and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103078.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103080.png" />-index of the semi-simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103081.png" />. The role of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103082.png" />-index is explained by the following theorem: Every semi-simple group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103083.png" /> is uniquely defined, up to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103084.png" />-isomorphism, by its class relative to an isomorphism over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103085.png" />, its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103086.png" />-index and its [[Anisotropic kernel|anisotropic kernel]]. The relative root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103087.png" /> is completely defined by the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103088.png" /> and by the set of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103090.png" /> (equal to 1 or 2), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103091.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103092.png" />. Conversely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103095.png" />, can be determined from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103096.png" />-index. In particular, two elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103097.png" /> have one and the same restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103098.png" /> if and only if they are located in the same orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r08103099.png" />; this defines a bijection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030100.png" /> and the set of orbits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030101.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030102.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030103.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030104.png" /> is the corresponding orbit, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030105.png" /> is any connected component in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030106.png" /> not all vertices of which lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030107.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030108.png" /> is the sum of the coefficients of the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030109.png" /> in the decomposition of the highest root of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030110.png" /> in simple roots.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081030/r081030112.png" />, then the above relative root system and relative Weyl group are naturally identified with the root system and Weyl group, respectively, of the corresponding symmetric space.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Tits,   "Sur la classification des groupes algébriques semi-simples"  ''C.R. Acad. Sci. Paris'' , '''249'''  (1959)  pp. 1438–1440</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Borel,   J. Tits,   "Groupes réductifs"  ''Publ. Math. IHES'' , '''27'''  (1965)  pp. 55–150</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Tits,   "Classification of algebraic simple groups" , ''Algebraic Groups and Discontinuous Subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc.  (1966)  pp. 33–62</TD></TR></table>