[[Zeta-function|zeta-function]] at the cost of introducing characters (cf.
...oup|Character of a group]]). The $L$-functions form a complicated class of special functions of a complex variable, defined by a
2 KB (347 words) - 21:23, 9 January 2015
The most important special cases are:
...tation theory is the study of the distribution of the irreducible ordinary characters of $ G $
6 KB (990 words) - 10:59, 29 May 2020
...r, who was motivated largely by a wish to obtain information about complex characters of finite groups (cf. [[Character of a group|Character of a group]]). One o
Such blocks have special significance; they are in bijection with the isomorphism types of projectiv
8 KB (1,167 words) - 06:46, 26 March 2023
...inatorics|Schur functions in algebraic combinatorics]]). In fact, both are special cases of Hall–Littlewood functions discovered by D.E. Littlewood [[#Refer
...l proof for the Murnaghan–Nakayama rule for computing the irreducible spin characters of $S _ { n }$ (cf. [[Representation of the symmetric groups|Representation
7 KB (917 words) - 20:49, 23 January 2024
...r representation|Regular representation]]). A monomial representation is a special case of an imprimitive representation (see [[Imprimitive group|Imprimitive
<TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Feit, "Characters of finite groups" , Benjamin (1967)</TD></TR>
4 KB (538 words) - 14:08, 17 March 2023
Usually, a distinction is made between general-purpose packages and special-purpose packages.
...plotting facilities, effective linear algebra, libraries with standard and special functions (e.g., trigonometric and Gamma-functions, Hermite polynomials) an
9 KB (1,292 words) - 20:51, 18 September 2016
...virtual character to mean a $\mathbb{Z}$-linear combination of irreducible characters). Such a group arises in the following way. Let $\mbf{G}$ denote a connecte
...centre of $\SL_n(q)$, is a finite group of Lie type called the projective special linear group. This group is a [[Simple finite group|finite simple group]] e
48 KB (8,458 words) - 18:22, 13 August 2023
with characters $ t _ {1} $,
...the case of semi-simple groups are measurable and locally integrable. The characters of irreducible unitary representations of solvable Lie groups of type I are
24 KB (3,516 words) - 08:27, 6 June 2020
The Schur functions $s_{ \lambda }$ are a special basis for the algebra of symmetric functions $\Lambda$. They are also intim
.../> is a class function). In fact, more is true: The irreducible polynomial characters of $\operatorname{GL}_l$ are precisely the $s_{ \lambda }$ for $\lambda$ wi
14 KB (2,001 words) - 10:09, 11 November 2023
The problem of studying these representations (computing their characters, finding explicit realizations, etc.) is the subject of an extensive theory
...op">[a4]</TD> <TD valign="top"> A. Wawrzyńczyk, "Group representations and special functions" , Reidel & PWN (1984) {{MR|0750113}} {{ZBL|0545.43001}} </TD
5 KB (643 words) - 00:34, 12 June 2021
The Chern–Simons functional is a special case of the Chern–Simons invariant and characteristic classes. General re
...top">[a3]</td> <td valign="top"> J. Cheeger, J. Simons, "Differential characters and geometric invariants" , ''Geometry and Topology (Maryland, 1983/4'' , '
4 KB (628 words) - 16:58, 1 July 2020
...erence set|Abelian difference set]]) has led to a satisfactory theory. The special case of cyclic difference sets is the one considered in [[Difference set|Di
...into an equation in the integral group ring ${\bf Z} G$; applying complex characters to the group ring element associated with $D$, this translates into the con
9 KB (1,331 words) - 19:36, 13 February 2024
...mpact group]]) and of L.S. Pontryagin [[#References|[2]]] on the theory of characters of locally compact Abelian groups (cf. [[Character of a group|Character of
...onic analysis on groups was developed mainly on the basis of the theory of characters of locally compact Abelian groups established by Pontryagin ([[#References|
66 KB (9,085 words) - 17:28, 31 March 2020
...for any partition $\{\lambda_1,\ldots,\lambda_p\}=\lambda$, and include as special cases the above functions, e.g. $S_{\{1,\ldots,1\}}=s_k$, $S_{\{ k \}}=p_k
||{{Ref|Li}}|| D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950)
5 KB (801 words) - 20:34, 13 September 2016
...d by the specialization $g = e$: $Z ( e , h ; z ) = T _ { h } ( z )$. Only special cases of these generalized moonshine conjectures have been proven.
...ra of a holomorphic $c = 24$ theory), in which context many of moonshine's special features appear natural [[#References|[a8]]], [[#References|[a17]]], and th
12 KB (1,765 words) - 09:46, 10 November 2023
...he derivation by the more general notion of an "intertwining mapping" . A special case of the automatic continuity problem for homomorphisms is the uniquenes
...tic functions of several complex variables, for the continuity of all such characters is given in [[#References|[a7]]].
11 KB (1,760 words) - 16:56, 1 July 2020
...solvable Lie groups [[#References|[5]]]. For certain orbits of the simple special group $ G _ {2} $
...condition is fulfilled, then the set of extensions is parametrized by the characters of the fundamental group of the orbit.
12 KB (1,682 words) - 08:04, 6 June 2020
...quence of the Peter–Weyl theorem is that the set of linear combinations of characters of the irreducible representations of $ G $
...>[a2]</TD> <TD valign="top"> A. Wawrzyńczyk, "Group representations and special functions" , Reidel (1984) pp. Sect. 4.4</TD></TR><TR><TD valign="top">[a
6 KB (855 words) - 16:40, 31 March 2020
...)$ of $ \operatorname {GL} _ { n } ( K )$ are in bijection with the linear characters of $T$. The module $\Delta ( \lambda )$ decomposes into direct sum of eigen
For further reference and results on the special case of general linear groups, in particular for explicit formulas for base
33 KB (5,081 words) - 10:26, 11 November 2023
...ot produce a resolvent of ${\frak G} _ { D }$; however, it does in certain special cases, e.g. when $D$ is analytical operator [[#References|[a6]]].
...ith the cohomology of the $G$-invariant Spencer complex if the non-trivial characters of $( G , G _ { 0 } )$ are non-characteristic.
8 KB (1,101 words) - 17:44, 1 July 2020