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[[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

The most important special cases are: ...tation theory is the study of the distribution of the irreducible ordinary characters of $ G $

...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

...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

...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>

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

...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

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

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

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 &amp; PWN (1984) {{MR|0750113}} {{ZBL|0545.43001}} </TD

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'' , '

...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

...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|

...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)

...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

...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]]].

...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.

...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

...)$ 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

...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.

is the set of all characters $ \alpha $ ...l group]] of the complex algebra $ \mathfrak g $[[#References|[7]]]. For characters in "general position" the representation $ e ( \alpha ) $

Let $\def\t{\tau}\t_b := \s_{n-m-b,n-m+1,\dots,n}$ be a special Schubert cycle (cf. ...e Giambelli formula for expressing an arbitrary Schubert cycle in terms of special Schubert cycles. Define $\t_b = 0$ if $B<0$ or $B>m$, and $\t_0 = 1$. Then

of rational characters of the torus $ T $ , in the special case of finite-dimensional semi-simple Lie algebras over the field of compl

in the sense of the Pontryagin theory of characters, when the compact group $ G ^ {*} $ ...ontryagin duality theorem (see [[Alexander duality|Alexander duality]]). A special case of these dualities is the Steenrod duality theorem (see [[Duality|Dual

...process|stochastic process]]) by a series or integral with respect to some special system of functions, such that the coefficients in this expansion are pairw to be some special space of entire analytic functions, then one arrives at a generalized stati

...tation|Unitary representation]]), which has many applications. A number of special properties simplify their study. In particular, the orthogonal complement t ...dying dual objects, and the problems of the theory of spherical functions, characters and harmonic analysis related to them, including the study of various group

...roups. A classification of reduced Abelian groups is known only in certain special cases. Thus, Ulm's theorem ([[#References|[1]]]) gives the classification o ...applied in many modern mathematical theories. Thus, the duality theory of characters for finite Abelian groups has been considerably extended to the duality the

....png" />. The arithmetic-mean geometric-mean inequality corresponds to the special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath. ...<TD valign="top"> R.A. Liebler, M.R. Vitale, "Ordering the partition characters of the symmetric group" ''J. of Algebra'' , '''25''' (1973) pp. 487–48

...hic form|Automorphic form]]) involve their Fourier coefficients. Here, the special case of holomorphic modular forms $f$ of weight $k$ for the full [[Modular ...tau ( p ) | \leq 2 p ^ { 11 / 2 }$. P. Deligne proved this conjecture as a special case of a much more general result holding for all cusp forms. This result

...ongruence on arbitrary semi-groups; 2) to describe all congruences on some special semi-groups, belonging to some class of importance. The first category incl ...mi-group of complex numbers, forms the subject of the theory of semi-group characters (cf. [[Character of a semi-group]]).

The special position of cyclotomic fields among all algebraic number fields is illustra runs through all non-trivial primitive multiplicative characters modulo $ n $.

...the functional equations of $L$-series associated to automorphic forms. In special settings, global root numbers are known to have deep connections to the van ...h, J. Queyrut, "On the functional equation of the Artin L-function for characters of real representations" ''Invent. Math.'' , '''20''' (1973) pp. 125–1

...f the letters of the Roman alphabet (which may be replaced or augmented by characters from a national set), digits, pairs of delimiters (parentheses), separators

...resentations of finite groups (for example, the orthogonality relation for characters, or for matrix entries), and also the [[Peter–Weyl theorem|Peter–Weyl t ...roperty have been studied; for example, the condition of finiteness of the special rank, different variants of the maximum and minimum conditions for subgroup

this has only been accomplished for special fields. The case when characters of a torus. The duality theorem states that the cup-product

To state Beilinson's conjectures on special values of $ L ( h ^ {i} ( X ) ,s ) $ Some special cases are as follows.

[[#References|[9]]] are applicable to special cases of such cohomology theories. ...opology duality manifest itself: in duality (in the sense of the theory of characters) between the homology and cohomology groups of the same dimension with dual

More special cases of bases of a set $ X $ ...nd, in particular, to employ the well-developed apparatus of the theory of characters.

...e of an $A$-module. The direct product and direct sum may be considered as special cases of the notions of a projective and an inductive limit. ...ntation]] of $G$ and are in one-to-one correspondence with the irreducible characters of the group. Modules over principal ideal rings and over Dedekind rings al

...ns $nk_l$, $(k,l)=1$, $n=0,1,\ldots,$ leads to the problem on the zeros of special zeta-functions — the so-called Dirichlet $L$-series, of the form ...re $a_n$ depends on $n$ and on the difference $k$ of the series (Dirichlet characters modulo $k$).

To do this he introduced certain arithmetical functions — characters $ \chi = \chi ( n, d) $( However, of special interest here are the results for values of $ d $

...order invariants expressed in terms of the elementary divisors of A — and characters of the form $ \chi ( q) = \pm 1 $. ...nces|[17]]]) in terms of generalized Gauss sums. Formula (3) includes as a special case Minkowski's formula for the weight of the genus:

...Galois group of $k^*$, $h$ the class number of $Q$ and $\chi_i$ the prime characters of $Q$, $1\leq i\leq h$. Then ...ever, at the time of writing (1978) the conjecture has been proved in very special cases only (rational surfaces, algebraic curves uniformizable by modular fu

An important special situation arises when the underlying ...ic model for growth, shape, reaction norms, and other infinite-dimensional characters. ''Journal of Mathematical Biology'' '''27''' 429--450.

...rzebruch's signature theorem (cf. also [[Signature|Signature]]) occupies a special place (see [[#References|[a33]]], especially for topics such as multiplicat ...$K$-theory of the algebra $A$ and the cyclic homology of $A$ is via Chern characters $\operatorname{Ch} : K _ { 0 } ( A ) \rightarrow \operatorname{HC} _ { 2 n