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mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations
Frobenius_formula
Mathematical problem
problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a mathematical problem that
Coin_problem
Theorem in linear algebra
matrix theory, the Perron–Frobenius theorem, proved in its first part by Oskar Perron (1907) and extended by Georg Frobenius (1912), asserts that a real
Perron–Frobenius_theorem
Mathematical formula for the number of Young tableaux
independently by Frobenius and Young in 1900 and 1902 respectively using algebraic methods. MacMahon found an alternate proof for the Young–Frobenius formula in 1916
Hook_length_formula
Process of extending a representation of a subgroup to the parent group
{f}}} is the unique map making the following diagram commute: The Frobenius formula states that if χ is the character of the representation σ, given by
Induced_representation
as Frobenius morphism, Frobenius map) Frobenius determinant theorem Frobenius formula Frobenius group Frobenius complement Frobenius kernel Frobenius inner
List of things named after Ferdinand Georg Frobenius
List_of_things_named_after_Ferdinand_Georg_Frobenius
Map raising elements to the pth power, in characteristic p
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with
Frobenius_endomorphism
Mathematical theorem
representation breaks up into discrete summands. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite
Selberg_trace_formula
mathematician Ferdinand Frobenius. Each covariant is a projection on the eigenspace associated with the eigenvalue λi. Frobenius covariants are the coefficients
Frobenius_covariant
direct sum of irreducible representations, and the trace formula is similar to the Frobenius formula for the character of the representation induced from
Arthur–Selberg_trace_formula
Norm on a vector space of matrices
circular shifts. The Frobenius norm is an extension of the Euclidean norm to K n × n {\displaystyle K^{n\times n}} and comes from the Frobenius inner product
Matrix_norm
Area of mathematics
{\displaystyle \chi _{\lambda }(C_{\rho })} can be computed using the Frobenius formula. The coefficients z ρ {\displaystyle z_{\rho }} are z ρ = ∏ j = 0
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Provides integral formulas for all derivatives of a holomorphic function
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a
Cauchy's_integral_formula
Numbers obtained by adding the two previous ones
Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio
Fibonacci_sequence
Concept in mathematical group theory
defining formula of Frobenius reciprocity can be extended to general complex-valued class functions. Given a matrix representation ρ of H, Frobenius later
Character_theory
Expresses the number of points of a variety over a finite field
Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its
Grothendieck_trace_formula
Operator encoding information about iterated map
Ruelle, or the Perron–Frobenius operator or Ruelle–Perron–Frobenius operator, in reference to the applicability of the Perron–Frobenius theorem to the determination
Transfer_operator
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Formula for number of orbits of a group action
1897 book on finite groups, attributing it to Frobenius 1887. But even prior to Frobenius, the formula was known to Cauchy in 1845. Consequently, this
Burnside's_lemma
Duality between the process of restricting and inducting in representation theory
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and
Frobenius_reciprocity
Concept in mathematics
{\displaystyle m_{\lambda }} may be computed by the Frobenius formula (or the hook length formula). For example, take n = 3 {\displaystyle n=3} . Then
Tensor product of representations
Tensor_product_of_representations
Formula in matrix theory
are the corresponding Frobenius covariants of A, which are (projection) matrix Lagrange polynomials of A. Sylvester's formula applies for any diagonalizable
Sylvester's_formula
Mapping theorem in topology
called Lefschetz trace formula that allows to express number of points of variety over finite field in terms of action of Frobenius morphism on its cohomologies
Lefschetz_fixed-point_theorem
Special kind of semigroup in mathematics
13}. The Frobenius number of S is F(S) = 13, and its conductor is 14. The genus of S : g(S) = 8. Numerical semigroups with small Frobenius number or
Numerical_semigroup
formula states: for a smooth algebraic stack X of finite type over a finite field F q {\displaystyle \mathbb {F} _{q}} and the "arithmetic" Frobenius
Behrend's_trace_formula
Polynomials used for interpolation
constant The Chebfun system Table of Newtonian series Frobenius covariant Sylvester's formula Finite difference coefficient Hermite interpolation Lagrange
Lagrange_polynomial
)} is called a Frobenius Lie algebra. If ( g , [ , ] , β ) {\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )} is a quasi-Frobenius Lie algebra, one
Quasi-Frobenius_Lie_algebra
German mathematician (1910–1990)
Collatz conjecture and remains unsolved. The Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a positive square matrix was also named after
Lothar_Collatz
Topics referred to by the same term
algebra) of the product of two matrices. Sylvester's closed solution for the Frobenius coin problem when there are only two coins. Sylvester's triangle problem
Sylvester's_theorem
Polynomial interpolation using derivative values
Neville's schema Bernstein polynomials Generalized Frobenius covariant Generalized Sylvester's formula Traub, J. F. (December 1964). "On Lagrange—Hermite
Hermite_interpolation
Mathematical conjecture about elliptic curves
of ap, the trace of Frobenius that appears in the formula. For the typical case (no complex multiplication, trace ≠ 0) their formula states that the number
Sato–Tate_conjecture
Mathematical expression
interpolation Carlson's theorem Table of Newtonian series Frobenius covariant Sylvester's formula Dunham, William (1990). "7". Journey Through Genius: The
Newton_polynomial
"Pushed forward" from one measurable space to another
transfer operator or Frobenius–Perron operator. In finite spaces this operator typically satisfies the requirements of the Frobenius–Perron theorem, and
Pushforward_measure
Distance function
n-dimensional open unit ball. Hilbert's metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces. Let Ω be a convex
Hilbert_metric
Cauchy's functional equation Cauchy filter Cauchy formula for repeated integration Cauchy–Frobenius lemma Cauchy identity Cauchy index Cauchy interlacing
List of things named after Augustin-Louis Cauchy
List_of_things_named_after_Augustin-Louis_Cauchy
Belgian mathematician
Deligne's contribution was to supply the estimate of the eigenvalues of the Frobenius endomorphism, considered the geometric analogue of the Riemann hypothesis
Pierre_Deligne
On generating functions from counting points on algebraic varieties over finite fields
value of any eigenvalue α of Frobenius on a fiber of E as follows. For any integer k, αk is an eigenvalue of Frobenius on a stalk of Ek, which for k
Weil_conjectures
Sum of elements on the main diagonal
B is a square matrix. The Frobenius inner product and norm arise frequently in matrix calculus and statistics. The Frobenius inner product may be extended
Trace_(linear_algebra)
the general theory.) It is a consequence of the Lefschetz trace formula for the Frobenius morphism that Z ( X , t ) = ∏ i = 0 2 dim X det ( 1 − t Frob
Local_zeta_function
Matrix operation generalizing exponentiation of scalar numbers
Phase-type distribution Lie product formula Baker–Campbell–Hausdorff formula Frobenius covariant Sylvester's formula Trigonometric functions of matrices
Matrix_exponential
Form of interpolation
For matrix arguments, this formula is called Sylvester's formula and the matrix-valued Lagrange polynomials are the Frobenius covariants. For a polynomial
Polynomial_interpolation
Algebraic structure
{\displaystyle \mathrm {GF} (p)} . It is called the Frobenius automorphism, after Ferdinand Georg Frobenius. Denoting by φk the composition of φ with itself
Finite_field
Figurate number
n} . Therefore, the triangular numbers may also be represented by the formula. T n = 1 + 2 + 3 + ⋯ + ( n − 1 ) + n = ∑ k = 1 n k . {\displaystyle T_{n}=1+2+3+\cdots
Triangular_number
Representation theory of finite groups Modular representation theory Frobenius reciprocity Restricted representation Induced representation Peter–Weyl
List of representation theory topics
List_of_representation_theory_topics
System of complete and orthogonal polynomials
singular points at x = ±1 so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for
Legendre_polynomials
Type of symmetric polynomials in mathematics
those for the hook partitions contained within the Young diagram. In Frobenius' notation, the partition is denoted ( a 1 , … , a r ∣ b 1 , … , b r )
Schur_polynomial
Concept in class field theory
distinguished generator is provided by the Frobenius automorphism. Certain conventions on terminology, such as arithmetic Frobenius, trace back to the fixing here
Weil_group
Solution to x*x + y*y + z*z = 3xyz
consists of three distinct integers. The unicity conjecture, as remarked by Frobenius in 1913, states that for a given Markov number c, there is exactly one
Markov_number
Aspect of algebraic number theory
The theory of the Frobenius element goes further, to identify an element of DPj / IPj for given j which corresponds to the Frobenius automorphism in the
Splitting of prime ideals in Galois extensions
Splitting_of_prime_ideals_in_Galois_extensions
In mathematics, invariant of square matrices
continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and
Determinant
Organic chemical compound
The name ether was given to the substance in 1729 by August Sigmund Frobenius. It was considered to be a sulfur compound until the idea was disproved
Diethyl_ether
German mathematician
"Orbifolding Frobenius Algebras". Internat. J. of Math. 14 (2003), 573-619 Kaufmann, Ralph M. "Singularities with Symmetries, Orbifold Frobenius algebras
Ralph_Kaufmann
Algebraic operation on coordinate vectors
_{r}=\int _{a}^{b}r(x)u(x)v(x)\,dx.} A double-dot product for matrices is the Frobenius inner product, which is analogous to the dot product on vectors. It is
Dot_product
Combinatorial object in representation theory
They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including
Young_tableau
Topics referred to by the same term
Quartic reciprocity Artin reciprocity Weil reciprocity for algebraic curves Frobenius reciprocity theorem for group representations Stanley's reciprocity theorem
Reciprocity_theorem
1175, ISSN 0021-8693, MR 1338967 Mehta, V. B.; Ramanathan, A. (1985), "Frobenius splitting and cohomology vanishing for Schubert varieties", Annals of
Demazure_module
Stochastic process
{\sqrt {2}}Im(A_{n-1,n}))} This is an isometry, where the matrix norm is Frobenius norm. By reversing this process, a standard Brownian motion in R n 2 {\textstyle
Dyson_Brownian_motion
Frobenius mappings, and under the modern understanding what is required for H is the transpose of Frobenius (see arithmetic and geometric Frobenius for
Hasse–Witt_matrix
Topic in mathematics
HS {\displaystyle \|\cdot \|_{\text{HS}}} is also called the Frobenius norm. Frobenius inner product – Binary operation, takes two matrices and returns
Hilbert–Schmidt_operator
Conjecture on zeros of the zeta function
function of a variety over a finite field correspond to eigenvalues of a Frobenius element on an étale cohomology group, the zeros of a Selberg zeta function
Riemann_hypothesis
German mathematician (1882–1965)
his PhD at the University of Berlin, where he was a student of Georg Frobenius, Hermann Schwarz and Issai Schur; his dissertation, Anwendung einer Formel
Ernst_Jacobsthal
named after Erik Fredholm List of things named after Ferdinand Georg Frobenius List of things named after Carl Friedrich Gauss List of things named after
Lists_of_mathematics_topics
Four-dimensional number system
)\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).} According to the Frobenius theorem, the algebra H {\displaystyle \mathbb {H} } is one of only two
Quaternion
English mathematician
complementing, and sometimes competing with, the work of Ferdinand Georg Frobenius, who began his research in the subject during the 1890s. One of Burnside's
William_Burnside
positive-semidefinite matrix Pfaffian Projection Spectral theorem Perron–Frobenius theorem List of matrices Diagonal matrix, main diagonal Diagonalizable
Outline_of_linear_algebra
Parameterization of a rotation into a unit vector and angle
case, the log is not unique. However, even in the case where θ = π the Frobenius norm of the log is ‖ log ( R ) ‖ F = 2 | θ | . {\displaystyle \|\log(R)\|_{\mathrm
Axis–angle_representation
polynomial or Gaussian coefficient Gauss transformation, also called Frobenius matrix Gauss–Bodenmiller theorem – described on website of University
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Matrix representing a Euclidean rotation
invariant under orthogonal transformations. A convenient choice is the Frobenius norm, ‖Q − M‖F, squared, which is the sum of the squares of the element
Rotation_matrix
Sorting algorithm which uses multiple comparison intervals
Shellsort is therefore connected with the Frobenius problem: for given integers h1,..., hn with gcd = 1, the Frobenius number g(h1,..., hn) is the greatest
Shellsort
German mathematician (1880–1975)
Perron effect Perron's formula Perron integral Perron method Perron number Perron tree Perron's irreducibility criterion Perron–Frobenius theorem Moessner's
Oskar_Perron
Method for producing composition algebras
finite-dimensional normed division algebras over the real numbers, while the Frobenius theorem states that the first three are the only finite-dimensional associative
Cayley–Dickson_construction
Counts pieces of a disk cut by lines
0) is given by the formula p = n 2 + n + 2 2 . {\displaystyle p={\frac {n^{2}+n+2}{2}}.} Using binomial coefficients, the formula can be expressed as
Lazy_caterer's_sequence
Theorem on polynomial roots modulo prime powers
Note that given an a ∈ F p {\displaystyle a\in \mathbb {F} _{p}} the Frobenius endomorphism y ↦ y p {\displaystyle y\mapsto y^{p}} gives a nonzero polynomial
Hensel's_lemma
{\displaystyle \phi \in \operatorname {Gal} (L/\mathbb {Q} )} be the Frobenius automorphism associated to β; the characteristic property of ϕ {\displaystyle
Proofs of quadratic reciprocity
Proofs_of_quadratic_reciprocity
Chicken nuggets sold by McDonald's
greatest number of McNuggets which cannot be purchased exactly is 43, the Frobenius number of the set {6,9,20}. This means that all natural numbers greater
Chicken_McNuggets
Polyhedral number representing a tetrahedron
20, 35, 56, 84, 120, 165, 220, ... (sequence A000292 in the OEIS) The formula for the nth tetrahedral number is represented by the 3rd rising factorial
Tetrahedral_number
'Best' approximation of a function by a rational function of given order
technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations
Padé_approximant
Most widely known generalized inverse of a matrix
converges to the matrix A {\displaystyle A} (in the maximum norm or Frobenius norm, say), then ( A n ) + {\displaystyle (A_{n})^{+}} need not converge
Moore–Penrose_inverse
Number that represents a hexagon with a dot in the center
631, 721, 817, 919. The nth centered hexagonal number is given by the formula H ( n ) = n 3 − ( n − 1 ) 3 = 3 n ( n − 1 ) + 1 = 3 n 2 − 3 n + 1. {\displaystyle
Centered_hexagonal_number
Concept in mathematics
bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it. Drinfeld modules were
Drinfeld_module
Indian mathematician
notion of Frobenius splitting of algebraic varieties jointly with Vikram Bhagvandas Mehta in (Mehta & Ramanathan 1985). The notion of Frobenius splitting
Annamalai_Ramanathan
Mathematic theory
p-power Frobenius acts semilinearly with respect to the Witt-vector Frobenius on K {\displaystyle K} . In the finite-field case, the q-power Frobenius acts
Monsky–Washnitzer_cohomology
Statistic comparing ordinal rankings
is the Frobenius inner product and ‖ A ‖ F = ⟨ A , A ⟩ F {\displaystyle \|A\|_{\rm {F}}={\sqrt {\langle A,A\rangle _{\rm {F}}}}} the Frobenius norm. In
Rank_correlation
Doubling map on the unit interval
called the transfer operator or the Ruelle–Frobenius–Perron operator. The largest eigenvalue is the Frobenius–Perron eigenvalue, and in this case, it is
Dyadic_transformation
Misunderstanding of the symbol of Hermes
context in the printer's device used by the Swiss medical printer Johann Frobenius (1460–1527), who depicted the staff entwined with serpents and surmounted
Caduceus as a symbol of medicine
Caduceus_as_a_symbol_of_medicine
Recursive integer sequence
immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. Another alternative expression
Catalan_number
Square matrices satisfy their characteristic equation
matrix of any degree". The general case was first proved by Ferdinand Frobenius in 1878. For a 1 × 1 {\displaystyle 1\times 1} matrix A = (a), the characteristic
Cayley–Hamilton_theorem
Period of the Fibonacci sequence modulo an integer
2 − x − 1 ) . {\displaystyle \mathbb {F} _{p}[x]/(x^{2}-x-1).} As the Frobenius automorphism x ↦ x p {\displaystyle x\mapsto x^{p}} exchanges these roots
Pisano_period
Arabic folk tale
Cave of the Wuarssen"), a variant from the Kabyle, was published by Leo Frobenius. An American variant was collected by Elsie Clews Parsons from Cape Verde
Ali Baba and the Forty Thieves
Ali_Baba_and_the_Forty_Thieves
Product of two prime numbers
cryptosystem. For a square semiprime n = p 2 {\displaystyle n=p^{2}} , the formula is again simple: φ ( n ) = p ( p − 1 ) = n − p . {\displaystyle \varphi
Semiprime
{\displaystyle \mathbb {C} } . ϕ {\displaystyle \phi } is the geometric Frobenius. # Bun G ( X ) ( F q ) = ∑ P 1 # Aut ( P ) {\displaystyle \#\operatorname
Moduli stack of principal bundles
Moduli_stack_of_principal_bundles
A prime p divides a^p–a for any integer a
generally used before starting a proof of primality. Fermat quotient Frobenius endomorphism p-derivation Fractions with prime denominators: numbers with
Fermat's_little_theorem
Aspect of mathematical group theory
fixing a point (which just correspond to inner automorphisms of S6). The Frobenius group of affine transformations of F5 (maps x ↦ a x + b {\displaystyle
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
Operation in group theory
the symmetric group on 3 symbols). This product is sometimes called the Frobenius product. In general, the product of two subgroups S and T is a subgroup
Product_of_group_subsets
Algebra describing 2D conformal symmetry
form at a given level N {\displaystyle N} is given by the Kac determinant formula, det ( S L , L ′ ( c , h ) ) L , L ′ ∈ L | L | = | L ′ | = N = A N ∏ 1
Virasoro_algebra
Type of magic based on imitation or correspondence
sanctuary of "timeless principle", were acted upon by rite. In 1933, Leo Frobenius, discussing cave paintings in North Africa, pointed out that many of the
Sympathetic_magic
Technique in mathematical group theory
Suppose that G is a reductive group defined over a finite field, with Frobenius map F. Ian G. Macdonald conjectured that there should be a map from general
Deligne–Lusztig_theory
groups Block (permutation group theory) Cayley's theorem Cycle index Frobenius group Galois group of a polynomial Jucys–Murphy element Landau's function
List_of_permutation_topics
American mathematician
group theory which had stood for around sixty years: the nilpotency of Frobenius kernels. At the time, this achievement was noted in The New York Times
John_G._Thompson
Ten raised to an integer power
name can be determined based on its Latin name-prefix using the following formula: 10[(prefix-number + 1) × 3] Examples: billion = 10[(2 + 1) × 3] = 109
Power_of_10
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