Gaussian binomial

In mathematics, the Gaussian binomials (sometimes called the Gaussian coefficients, or the "q"-binomial coefficients) are the q-analogs of the binomial coefficients.

Definition

The Gaussian binomials are defined by

:$\{m\; choose\; r\}\_q=\; frac\{(1-q^m)(1-q^\{m-1\})cdots(1-q^\{m-r+1\})\}\; \{(1-q)(1-q^2)cdots(1-q^r)\}.$

One can prove that

:$lim\_\{q\; ightarrow\; 1^-\}\; \{m\; choose\; r\}\_q\; =\; \{m\; choose\; r\}.$

The Pascal identities for the Gaussian binomials are

:$\{m\; choose\; r\}\_q\; =\; q^r\; \{m-1\; choose\; r\}\_q\; +\; \{m-1\; choose\; r-1\}\_q$

and

:$\{m\; choose\; r\}\_q\; =\; \{m-1\; choose\; r\}\_q\; +\; q^\{m-r\}\{m-1\; choose\; r-1\}\_q.$

The Newton binomial formulas are

:$prod\_\{k=0\}^\{n-1\}\; (1+q^kt)=sum\_\{k=0\}^n\; q^\{k(k-1)/2\}\; \{n\; choose\; k\}\_q\; t^k$

and

:$prod\_\{k=0\}^\{n-1\}\; frac\{1\}\{(1-q^kt)\}=sum\_\{k=0\}^infty\; \{n+k-1\; choose\; k\}\_q\; t^k.$

Like the ordinary binomial coefficients, the Gaussian binomials are center-symmetric i.e. invariant under the reflection $r\; ightarrow\; m-r$:

:$\{m\; choose\; r\}\_q\; =\; \{m\; choose\; m-r\}\_q.$

The first Pascal identity allows one to compute the Gaussian binomials recursively (with respect to "m" ) using the initial "boundary" values

:$\{m\; choose\; m\}\_q\; =\{m\; choose\; 0\}\_q=1$

and also incidentally shows that the Gaussian binomials are indeed polynomials (in "q"). The second Pascal identity follows from the first using the substitution $r\; ightarrow\; m-r$ and the invariance of the Gaussian binomials under the reflection $r\; ightarrow\; m-r$. Both Pascal identities together imply

:$\{m\; choose\; r\}\_q\; =$1-q^{mover {1-q^{m-r} {m-1 choose r}_q

which leads (when applied iteratively for "m", "m" − 1, "m" − 2,....) to an expression for the Gaussian binomial as given in the definition above.

Applications

Gaussian binomials occur in the counting of symmetric polynomials and in the theory of partitions. The coefficient of "q"^"r" in

:$\{n+m\; choose\; m\}\_q$

is the number of partitions of "r" with "m" or fewer parts each less than or equal to "n". Equivalently, it is also the number of partitions of "r" with "n" or fewer parts each less than or equal to "m".

Gaussian binomials also play an important role in the enumerative theory of symmetric spaces defined over a finite field. In particular, for every finite field "F"_"q" with "q" elements, the Gaussian binomial

:$\{n\; choose\; k\}\_q$

counts the number "v"_"n","k";"q" of different "k"-dimensional vector subspaces of an "n"-dimensional vector space over "F"_"q" (a Grassmannian). For example, the Gaussian binomial

:$\{n\; choose\; 1\}\_q=1+q+q^2+cdots+q^\{n-1\}$

is the number of different lines in "F"_"q"^"n" (a projective space).

In the conventions common in applications to quantum groups, a slightly different definition is used; the quantum binomial there is:$q^\{k^2\; -\; n\; k\}\{n\; choose\; k\}\_\{q^2\}$.This version of the quantum binomial is symmetric under exchange of $q$ and $q^\{-1\}$.

References

* Eugene Mukhin, [http://mathcircle.berkeley.edu/BMC3/SymPol.pdf Symmetric Polynomials and Partitions] (undated, 2004 or earlier).
* Ratnadha Kolhatkar, [http://www.math.mcgill.ca/goren/SeminarOnCohomology/GrassmannVarieties%20.pdf Zeta function of Grassmann Varietes] (dated January 26, 2004)
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