Examples of "monomial"
An induced representation of a one dimensional representation is called a monomial representation, because it can be represented as monomial matrices. Some groups have the property that all of their irreducible representations are monomial, the so-called monomial groups.
In this section only finite groups are considered. A monomial group is solvable by , presented in textbook in and . Every supersolvable group and every solvable A-group is a monomial group. Factor groups of monomial groups are monomial, but subgroups need not be, since every finite solvable group can be embedded in a monomial group, as shown by and in textbook form in .
as a linear combination of monomials, where is a finite subset of and the are all nonzero. When a monomial order has been chosen, the leading monomial is the largest in , the leading coefficient is the corresponding , and the leading term is the corresponding . "Head" monomial/coefficient/term is sometimes used as a synonym of "leading". Some authors use "monomial" instead of "term" and "power product" instead of "monomial". In this article, a monomial is assumed to not include a coefficient.
Monomial matrices occur in representation theory in the context of monomial representations. A monomial representation of a group "G" is a linear representation "ρ" : "G" → GL("n", "F") of "G" (here "F" is the defining field of the representation) such that the image "ρ"("G") is a subgroup of the group of monomial matrices.
Furthermore, if formula_31 is the highest-degree monomial term in formula_20, then the highest-degree monomial term in formula_33 is formula_34. Consequently, the polynomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
Although Gröbner basis theory does not depend on a particular choice of an admissible monomial ordering, three monomial orderings are specially important for the applications:
Let us consider a polynomial ring formula_32 in which the variables are split into two subsets "X" and "Y". Let us also choose an elimination monomial ordering "eliminating" "X", that is a monomial ordering for which two monomials are compared by comparing first the "X"-parts, and, in case of equality only, considering the "Y"-parts. This implies that a monomial containing an "X"-variable is greater than every monomial independent of "X".
monomial symmetric polynomials of degree in , for instance
Since these conditions may be easier to verify for a monomial order defined through an explicit rule, than to directly prove it is a well-ordering, they are sometimes preferred in definitions of monomial order.
Although the dimension and the degree do not depend on the choice of the monomial ordering, the Hilbert series and the polynomial formula_29 change when one changes of monomial ordering.
This law of composition on formula_119 causes the "monomial representation"
triangularity property when expanded in the monomial basis.
The inverse of a non-negative matrix is usually not non-negative. The exception is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus the inverse of a positive matrix is not positive or even non-negative, as positive matrices are not monomial, for dimension formula_2
is called the monomial representation of formula_1 in formula_119
similar to the "permutation representation" and the "monomial representation".
These monomial symmetric polynomials form a vector space basis: every symmetric polynomial "P" can be written as a linear combination of the monomial symmetric polynomials. To do this it suffices to separate the different types of monomial occurring in "P". In particular if "P" has integer coefficients, then so will the linear combination.
Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the prefix "bi" (two in Latin), a monomial should theoretically be called a "mononomial". "Monomial" is a syncope by haplology of "mononomial".
In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1 .
A useful property of the degree reverse lexicographical order is that a homogeneous polynomial is a multiple of the least indeterminate if and only if its leading monomial (its greater monomial) is a multiple of this least indeterminate.