Quasideterminant

In mathematics, the quasideterminant is a replacement for the determinant for matrices 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 [1][2] and Heyting, [3] 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 and V.S. Retakh. [4][5] 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.

DefinitionEdit

 
(a picture 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.

IdentitiesEdit

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 decomposition of an   matrix   with   a   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 [4][5] 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 formula.

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

 

or

 

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, [6] 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 determinantsEdit

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

ApplicationsEdit

Paraphrasing their 2005 survey article with S. Gelfand and R. Wilson ,[7] 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." Substantive use has been made of the quasideterminant in such fields of mathematics as integrable systems, [8][9] representation theory, [10][11] algebraic combinatorics, [12] the theory of noncommutative symmetric functions, [13] the theory of polynomials over division rings, [14] and noncommutative geometry. [15][16][17]

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 do Grassmannians and flags over commutative fields. More information on these can be found in the survey article.[7]

See alsoEdit

ReferencesEdit

  1. ^ A.R. Richardson, Hypercomplex determinants, Messenger of Math. 55 (1926), no. 1.
  2. ^ A.R. Richardson, Simultaneous linear equations over a division algebra, Proc. London Math. Soc. 28 (1928), no. 2.
  3. ^ A. Heyting, Die theorie der linearen gleichungen in einer zahlenspezies mit nichtkommutativer multiplikation, Math. Ann. 98 (1928), no. 1.
  4. ^ a b I. Gelfand, V. Retakh, Determinants of matrices over noncommutative rings, Funct. Anal. Appl. 25 (1991), no. 2.
  5. ^ a b I. Gelfand, V. Retakh, Theory of noncommutative determinants, and characteristic functions of graphs, Funct. Anal. Appl. 26 (1992), no. 4.
  6. ^ D. Krob, B. Leclerc, Minor identities for quasi-determinants and quantum determinants, Comm. Math. Phys. 169 (1995), no. 1.
  7. ^ a b I. Gelfand, S. Gelfand, V. Retakh, R.L. Wilson, Quasideterminants. Adv. Math. 193 (2005), no. 1. (eprint)
  8. ^ P. Etingof, I. Gelfand, V. Retakh, Nonabelian integrable systems, quasideterminants, and Marchenko lemma. Math. Res. Lett. 5 (1998), no. 1-2.
  9. ^ C.R. Gilson, J.J.C. Nimmo, C.M. Sooman, On a direct approach to quasideterminant solutions of a noncommutative modified KP equation, J. Phys. A: Math. Theor. 41 (2008), no. 8. (eprint)
  10. ^ A. Molev, Yangians and their applications, in Handbook of algebra, Vol. 3, North-Holland, Amsterdam, 2003. (eprint)
  11. ^ J. Brundan, A. Kleshchev, Parabolic presentations of the Yangian Y(gl_n), Comm. Math. Phys. 254 (2005). (eprint)
  12. ^ M. Konvalinka, I. Pak, Non-commutative extensions of the MacMahon Master Theorem, Adv. Math. 216 (2007), no. 1. (eprint)
  13. ^ I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, J.-Y. Thibon, Noncommutative symmetric functions. Adv. Math. 112 (1995), no. 2. (eprint)
  14. ^ I. Gelfand, V. Retakh, Noncommutative Vieta theorem and symmetric functions. The Gelfand Mathematical Seminars, 1993--1995.
  15. ^ Z. Škoda, Noncommutative localization in noncommutative geometry, in "Non-commutative localization in algebra and topology", London Math. Soc. Lecture Note Ser., 330, Cambridge Univ. Press, Cambridge, 2006. (eprint)
  16. ^ A. Lauve, Quantum and quasi-Plücker coordinates, J. Algebra (296) 2006, no. 2. (eprint)
  17. ^ A. Berenstein, V. Retakh, Noncommutative double Bruhat cells and their factorizations, IMRN 2005. (eprint)