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What (who) is Hermitian-symmetric - definition

MAP TAKING TWO VECTORS FROM A COMPLEX VECTOR SPACE AND RETURNING A COMPLEX NUMBER, WHICH IS LINEAR IN ONE VARIABLE AND SEMILINEAR IN ANOTHER VARIABLE

Sesquilinear; Hermitian form; Skew-Hermitian form; Hermitian space; Hermitian product; Semi-bilinear form; Symmetric sesquilinear form; Antisymmetric sesquilinear form

Hermitian symmetric space

MANIFOLD WITH INVERSION SYMMETRY

Hermitean symmetric space; Bounded symmetric domain; Hermitian symmetric domain; Cartan domain

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.

https://en.wikipedia.org/wiki/Hermitian_symmetric_space

Hermitian Yang–Mills connection

A HERMITIAN HOLOMORPHIC VECTOR BUNDLE OVER A KÄHLER MANIFOLD, WHOSE CHERN CONNECTION’S CURVATURE SATISFIES EINSTEIN’S EQUATIONS (I.E. EQUALS THE IDENTITY TIMES A CONSTANT)

Hermitian–Einstein metric; Einstein-Hermitian vector bundle; Hermitian-Einstein vector bundle; Hermitian–Einstein vector bundle; Einstein–Hermitian metric; Einstein-Hermitian metric; Hermitian-Einstein metric; Hermitian-Einstein connection; Hermitian–Einstein connection; Einstein-Hermitian connection; Einstein–Hermitian connection; Einstein–Hermitian vector bundle; Hermite-Einstein connection; Hermite–Einstein connection; Hermite-Einstein vector bundle; Hermite-Einstein metric; Hermitian Yang-Mills equations; Hermitian Yang-Mills connection; Hermitian Yang–Mills equation; Hermitian Yang-Mills equation; Hermitian Yang–Mills equations; Hermite–Einstein metric

In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction of the curvature 2-form of the connection with the Kähler form is required to be a constant times the identity transformation. Hermitian Yang–Mills connections are special examples of Yang–Mills connections, and are often called instantons.

https://en.wikipedia.org/wiki/Hermitian_Yang–Mills_connection

Elementary symmetric polynomial

HOMOGENEOUS SYMMETRIC POLYNOMIAL IN WHICH EACH POSSIBLE MONOMIAL OCCURS EXACTLY ONCE WITH COEFFICIENT 1

Elementary symmetric function; Elementary symmetric polynomials; Fundamental theorem of symmetric polynomials; Fundamental Theorem of Symmetric Polynomials

In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials.

https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial

Wikipedia

Sesquilinear form

In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector.

A motivating special case is a sesquilinear form on a complex vector space, V. This is a map V × V → C that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any field and the twist is provided by a field automorphism.

An application in projective geometry requires that the scalars come from a division ring (skew field), K, and this means that the "vectors" should be replaced by elements of a K-module. In a very general setting, sesquilinear forms can be defined over R-modules for arbitrary rings R.

https://en.wikipedia.org/wiki/Sesquilinear form