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The number of these irreducibles is equal to the number of conjugacy classes of G. The above fact can be explained by character theory. Recall that the character of the regular representation χ(g) is the number of fixed points of g acting on the regular representation V. It means the number of fixed points χ(g) is zero when g is not id and |G ...
The condensation of a directed graph G is a directed acyclic graph with one vertex for each strongly connected component of G, and an edge connecting pairs of components that contain the two endpoints of at least one edge in G. cone A graph that contains a universal vertex. connect Cause to be connected. connected
1. Inner semidirect product: if N and H are subgroups of a group G, such that N is a normal subgroup of G, then = and = mean that G is the semidirect product of N and H, that is, that every element of G can be uniquely decomposed as the product of an element of N and an element of H.
The Littlewood–Richardson rule states that , is equal to the number of Littlewood–Richardson tableaux of skew shape / and of weight . Pieri's formula is a special case of the Littlewood-Richardson rule, which expresses the product h r s λ {\displaystyle h_{r}s_{\lambda }} in terms of Schur polynomials.
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows. The objects are the elements of G , and for any two elements x and y of G , there exists a unique morphism from x to y if and only if x ∼ y . {\displaystyle x\sim y.}
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
In Boolean logic, the majority function (also called the median operator) is the Boolean function that evaluates to false when half or more arguments are false and true otherwise, i.e. the value of the function equals the value of the majority of the inputs.
A formal expression is a kind of string of symbols, created by the same production rules as standard expressions, however, they are used without regard to the meaning of the expression. In this way, two formal expressions are considered equal only if they are syntactically equal, that is, if they are the exact same expression.