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The quantum binomial expansion for $q$ -commuting elements

Statement

Let $t$ be an indeterminate over $Q$ , and let $A$ be a unital associative $Q (t)$ -algebra. For $N \geq 0$ and $0 \leq r \leq N$ , set

$B_{N, r} (t) : = \frac{[N]_{t}!}{[r]_{t}! [N - r]_{t}!}, [m]_{t} : = 1 + t + \dots + t^{m - 1}, [m]_{t}! : = \prod_{k = 1}^{m} [k]_{t}, [0]_{t} : = 0, [0]_{t}! : = 1,$

and set $B_{N, r} (t) = 0$ when $r < 0$ or $r > N$ . If $x, y \in A$ satisfy $y x = t x y$ , then for every $N \geq 0$ ,

$(x + y)^{N} = \sum_{r = 0}^{N} B_{N, r} (t) x^{r} y^{N - r} .$

For a symmetrizable Cartan datum and $i \in I$ , any unital $Q (q)$ -algebra can be viewed as a $Q (t)$ -algebra via $t \mapsto q_{i}^{2}$ . In that algebra, if $y x = q_{i}^{2} x y$ , then

$(x + y)^{N} = \sum_{r = 0}^{N} q_{i}^{r (N - r)} {(\binom{N}{r})}_{i} x^{r} y^{N - r},$

using the symmetric Gaussian binomials of Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ . If instead $y x = t^{- 1} x y$ , the same expansion has coefficients $B_{N, r} (t^{- 1}) = t^{- r (N - r)} B_{N, r} (t)$ .

Facts & Assumptions

Given: The conventions above, the Gaussian quotient definitions, and the relation $y x = t x y$ when the generic expansion is used.

[F1]

The asymmetric $q$ -integer and factorial use $[m]_{t} = 1 + t + \dots + t^{m - 1}$ and the empty product $[0]_{t}! = 1$ ; for $k \geq 1$ , $[k]_{t}$ is a nonzero polynomial, so the Gaussian factorial quotient is defined in $Q (t)$ (The $q$ -integer, $q$ -factorial and $q$ -multinomial coefficients).

[F2]

The symmetric Gaussian coefficient satisfies $C_{N, r} = q_{i}^{- r} C_{N - 1, r} + q_{i}^{N - r} C_{N - 1, r - 1}$ (The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality).

[F3]

The symmetric and asymmetric Gaussian coefficients satisfy ${(\binom{N}{r})}_{i} = q_{i}^{- r (N - r)} B_{N, r} (q_{i}^{2})$ , with $q_{i} = q^{d_{i}}$ and $d_{i} > 0$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F4]

Since $q$ is indeterminate and $d_{i} > 0$ , $q_{i}^{2} = q^{2 d_{i}}$ is transcendental; substitution $t \mapsto q_{i}^{2}$ therefore embeds $Q (t)$ into $Q (q)$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

Proof

technique · Induct on the power, using the exact q-Pascal coefficient for the chosen commutation order

1.1givenF1algebra

For $k \geq 1$ , $[k]_{t^{- 1}} = t^{- (k - 1)} [k]_{t}$ ; multiplying for $k = 1, \dots, N$ and dividing the factorials gives $B_{N, r} (t^{- 1}) = t^{- N (N - 1) / 2 + r (r - 1) / 2 + (N - r) (N - r - 1) / 2} B_{N, r} (t) = t^{- r (N - r)} B_{N, r} (t)$ . The exponent equality follows by expanding the three quadratic terms; for $N = 0, r = 0$ both coefficients equal $1$ .

1.2givenF2F3F4algebra

Put $D_{N, r} : = q_{i}^{r (N - r)} C_{N, r}$ . Multiplying the recurrence [F2] by $q_{i}^{r (N - r)}$ and using [F3] gives $B_{N, r} (q_{i}^{2}) = B_{N - 1, r} (q_{i}^{2}) + (q_{i}^{2})^{N - r} B_{N - 1, r - 1} (q_{i}^{2})$ for $0 \leq r \leq N$ ; outside this range all terms vanish. The first exponent becomes $r (N - r - 1)$ , and the second differs from $(r - 1) (N - r)$ by $2 (N - r)$ . By the injectivity in [F4], this is the generic recurrence $B_{N, r} (t) = B_{N - 1, r} (t) + t^{N - r} B_{N - 1, r - 1} (t)$ .

2.1step 1.2F3algebra

For $N = 0$ the formula is $1 = 1$ . Suppose it holds for $N - 1$ . From $y x = t x y$ , induction on $a$ gives $y^{a} x = t^{a} x y^{a}$ : it is true for $a = 0$ , and $y^{a + 1} x = t^{a} y x y^{a} = t^{a + 1} x y^{a + 1}$ . Multiplying the $N - 1$ expansion on the right by $x + y$ and reindexing the terms from the final $x$ gives the coefficient $B_{N - 1, r} (t) + t^{N - r} B_{N - 1, r - 1} (t)$ at $x^{r} y^{N - r}$ . By step 1.2 this is $B_{N, r} (t)$ , proving the generic expansion. Under $t = q_{i}^{2}$ , [F3] turns this coefficient into $q_{i}^{r (N - r)} C_{N, r}$ and gives the symmetric formula.

3.1step 1.1step 2.1algebra∎

If $y x = t^{- 1} x y$ , apply the generic expansion of step 2.1 with parameter $t^{- 1}$ . Step 1.1 rewrites its coefficients as $t^{- r (N - r)} B_{N, r} (t)$ , proving the inverse-parameter formula as well.

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Sources

Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (book-length lecture notes, last updated 18 January 2024) (standard reference, not scraped)
Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)

The quantum binomial expansion for qq-commuting elements