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Group representation
A group representation, the subject of group representation theory, realizes an abstract group as invertible linear transformations of a vector space, with applications in chemistry, physics, and quantum mechanics.
Group ring
A group ring R[G] is a ring of formal finite sums of group elements with coefficients in a ring R; when R is a field it is a group algebra.
Group scheme
A group scheme is a scheme with a group structure given by morphisms for multiplication, inverse, and identity, so its points form groups for every base scheme.
Group theory
Group theory is the branch of mathematics studying groups, algebraic structures that capture symmetry and operations like addition, founded in Galois's 1830s work on equations.
Groupoid
A groupoid is a small category in which every morphism is an isomorphism, equivalently a set with a partial multiplication, introduced by Heinrich Brandt in 1926.
Hadamard matrix
A Hadamard matrix is a square matrix whose entries are all +1 or −1 and whose rows are mutually orthogonal, achieving the largest determinant possible of n^(n/2).
Hadamard product (matrices)
The Hadamard product, also called the Schur product or entrywise product, multiplies two matrices of the same size entry by entry, unlike ordinary matrix multiplication.
Half-precision floating-point format
Half-precision floating-point format, also called FP16 or binary16, is a 16-bit IEEE 754 encoding that stores numbers in half the memory of single precision, widely used in graphics and machine learning.
Hamming bound
The Hamming bound, also called the sphere-packing bound, is an upper limit on the size of an error-correcting code with given length and minimum distance.
Hankel matrix
A Hankel matrix is a square or rectangular matrix whose entries are constant along each anti-diagonal, named after the German mathematician Hermann Hankel.
Happy number
A happy number is a positive integer that reaches 1 when repeatedly replaced by the sum of the squares of its digits, for example 7, 13, and 19.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is an analytic number theory technique for counting integer representations as sums of powers or primes, arising from Hardy and Ramanujan's 1917 partition work.
Hardy–Littlewood conjecture
The Hardy–Littlewood conjecture, made by G. H. Hardy and John E. Littlewood in 1923, predicts how often patterns of primes occur and generalizes the twin prime conjecture.
Harish-Chandra isomorphism
The Harish-Chandra isomorphism is an algebra isomorphism identifying the center of the universal enveloping algebra of a semisimple Lie algebra with Weyl-group-invariant polynomials on a Cartan subalgebra.
Harmonic number
A harmonic number is the sum of the reciprocals of the first n positive integers, a partial sum of the divergent harmonic series that grows like the natural logarithm.
Harmonic series (mathematics)
The harmonic series is the infinite sum of 1 + 1/2 + 1/3 + 1/4 + …, which diverges even though its terms tend to zero.
Harshad number
A Harshad number, also called a Niven number, is a positive integer divisible by the sum of its digits; 1729 qualifies because 1729 = 91 × 19, the sum of its digits.
Hasse diagram
A Hasse diagram is a drawing of a finite partially ordered set, showing an edge between two elements exactly when one covers the other, with greater elements higher.
Hasse norm theorem
The Hasse norm theorem, Hasse's theorem on cyclic extensions, is a 1931 result of Helmut Hasse: in a cyclic extension of number fields, being a norm locally everywhere is equivalent to being a global norm.
Hasse principle
The Hasse principle, also called the local-global principle, is a statement in number theory about when solutions to equations over the rational numbers follow from local solutions.
Hecke algebra
A Hecke algebra is a commutative algebra of operators acting on modular forms, generated by the Hecke operators T_n, which encode averaging over sublattices of a lattice.
Hecke operator
A Hecke operator is a linear operator on spaces of modular forms, introduced by Erich Hecke in 1937, whose commuting eigenvalues decompose them into eigenforms.
Heegner number
A Heegner number is a squarefree positive integer for which the imaginary quadratic field Q(√−n) has class number one; there are exactly nine: 1, 2, 3, 7, 11, 19, 43, 67, and 163.
Height function
A height function in Diophantine geometry measures the arithmetic complexity of rational points on a projective variety, and underlies finiteness results such as the Mordell–Weil theorem.
Heisenberg algebra
The Heisenberg algebra is a nilpotent Lie algebra, formally the Heisenberg Lie algebra, whose brackets encode the canonical commutation relations of quantum mechanics; Hermann Weyl identified it in the 1920s.
Heisenberg group
The Heisenberg group is a nilpotent Lie group of real upper triangular 3 × 3 matrices with ones on the diagonal, named for its role in quantum mechanics.
Held group
The Held group (He) is a sporadic finite simple group discovered by Dieter Held in 1969 while studying groups related to the Mathieu group M24.
Hensel's lemma
Hensel's lemma, or Hensel lifting, is a number theory theorem stating that a simple root of a polynomial modulo a prime p lifts uniquely to a p-adic root.
Henselian ring
A henselian ring, also called a Hensel local ring, is a local ring in which simple roots of polynomials over the residue field lift to roots in the ring.
Hermite normal form
The Hermite normal form (HNF) is a unique canonical echelon-like form of an integer matrix, reached by multiplying by a matrix of determinant ±1 while preserving the generated lattice.