Quasideterminant - AbsoluteAstronomy.com

In mathematics, the quasideterminant is a replacement for the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

for matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

with noncommutative entries. Example 2 × 2 quasideterminants are as follows:

In general, there are n² quasideterminants defined for an n × n matrix (one for each position in the matrix), but the presence of the inverted terms above should give the reader pause: they are not always defined, and even when they are defined, they do not reduce to determinants when the entries commute. Rather,

where

means delete the ith row and jth column from A.

The

examples above were introduced between 1926 and 1928 by Richardson

Archibald Read Richardson

Archibald Read Richardson was a British mathematician known for his work in algebra.He collaborated with Dudley E. Littlewood on invariants and group representation theory...

and Heyting,

but they were marginalized at the time because they were not polynomials in the entries of

. These examples were rediscovered and given new life in 1991 by I.M. Gelfand

Israel Gelfand

Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis...

and V.S. Retakh.
There, they develop quasideterminantal versions of many familiar determinantal properties. For example, if

is built from

by rescaling its

-th row (on the left) by

, then

.
Similarly, if

is built from

by adding a (left) multiple of the

-th row to another row, then

. They even develop a quasideterminantal
version of Cramer's rule

Cramer's rule

In linear algebra, Cramer's rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution...

Definition

Let

be an

matrix over a (not necessarily commutative)
ring

and fix

. Let

denote the (

)-entry of

, let

denote the

-th row of

with column

deleted, and let

denote the

-th column of

with row

deleted. The (

)-quasideterminant of

is defined if the submatrix

is invertible over

. In this case,

Recall the formula (for commutative rings) relating

to the determinant, namely

. The above definition is a generalization in that (even for noncommutative rings) one has

whenever the two sides makes sense.

Identities

One of the most important properties of the quasideterminant is what Gelfand and Retakh
call the “heredity principle.” It allows one to take a quasideterminant in
stages (and has no commutative counterpart). To illustrate, suppose

is a block matrix

Block matrix

In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...

decomposition of an

matrix

with

matrix. If the (

)-entry of

lies within

, it says that

That is, the quasideterminant of a quasideterminant is a quasideterminant! To put it less succinctly: UNLIKE determinants, quasideterminants treat matrices with block-matrix entries no differently than ordinary matrices (something determinants cannot do since block-matrices generally don't commute with one another). That is, while the precise form of the above identity is quite surprising, the existence of some such identity is less so.
Other identities from the papers are (i) the so-called “homological relations,” stating that two quasideterminants in a common row or column are closely related to one another, and (ii) the Sylvester

James Joseph Sylvester

James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...

formula.

(i) Two quasideterminants sharing a common row or column satisfy

respectively, for all choices

so that the
quasideterminants involved are defined.

(ii) Like the heredity principle, the Sylvester identity is a way to recursively
compute a quasideterminant. To ease notation, we display a special case. Let

be the upper-left

submatrix of an

matrix

and fix a coordinate (

) in

. Let

be the

matrix, with

defined as the (

)-quasideterminant of the

matrix formed by adjoining to

the first

columns of row

, the first

rows of column

, and the entry

. Then one has

Many more identities have appeared since the first articles of Gelfand and Retakh on the subject, most of them being analogs of classical determinantal identities. An important source is Krob and Leclerc's 1995 article,
To highlight one, we consider the row/column expansion identities. Fix a row

to expand along. Recall the determinantal formula

.
Well, it happens that quasideterminants satisfy

(expansion along column

), and

(expansion along row

Connections to other determinants

The quasideterminant is certainly not the only existing determinant analog for noncommutative settings—perhaps the most famous examples are the Dieudonné determinant

Dieudonné determinant

In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings.It was introduced by ....

and quantum

Quantum group

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...

determinant. However, these are related to the quasideterminant in some way. For example,

with the factors on the right-hand side commuting with each other. Other famous examples, such as Berezinian

Berezinian

In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin...

s, Moore and Study determinants, Capelli determinants, and Cartier-Foata-type determinants are also expressible in terms of quasideterminants. Gelfand has been known to define a (noncommutative) determinant as “good” if it may be expressed as products of quasiminors.

Applications

Paraphrasing their 2005 survey article with S. Gelfand and R. Wilson
,
Gelfand and Retakh advocate for the adoption of
quasideterminants as “a main organizing tool in noncommutative algebra, giving
them the same role determinants play in commutative algebra.” By now,
substantive use has been made of the quasideterminant in such fields of mathematics as
integrable systems,
representation theory,
algebraic combinatorics,

the theory of noncommutative symmetric functions,

the theory of polynomials over division rings,

and noncommutative geometry.
Several of the applications above make use of quasi-Plücker coordinates, which parametrize noncommutative Grassmannians and flags in much the same way as Plücker coordinates

Plücker embedding

In mathematics, the Plücker embedding describes a method to realize the Grassmannian of all r-dimensional subspaces of a vector space V as a subvariety of the projective space of the rth exterior power of that vector space, P....

do Grassmannians and flags over commutative fields. More information on these can be found in the survey article.

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