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Integrality of the string decomposition and the divided-power string ideals

Statement

Let $i \in I$ and let $e_{i}^{'}$ be the skew derivation of Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half, with string decomposition $u = \sum_{l} F_{i}^{(l)} u_{l}$ , $u_{l} \in \ker e_{i}^{'}$ (The $i$ -string decomposition of the negative half).

(i) If $u \in U_{A}^{-}$ then all $u_{l}$ lie in $U_{A}^{-}$ . The integral projector is $P_{i} = \sum_{l} (- 1)^{l} q_{i}^{- l (l - 1) / 2} F_{i}^{(l)} (e_{i}^{'})^{l}, u_{l} = q_{i}^{l (l - 1) / 2} P_{i} (e_{i}^{'})^{l} u,$ where all powers of $e_{i}^{'}$ are ordinary powers, not divided by $[l]_{i}!$ .

(ii) Define $I_{(i, n)}^{A} : = U_{A}^{-} \cap F_{i}^{n} U_{q}^{-} = ⨁_{l \geq n} F_{i}^{(l)} (\ker e_{i}^{'} \cap U_{A}^{-}) = \sum_{l \geq n} F_{i}^{(l)} U_{A}^{-},$ $J_{(i, n)}^{A} : = π_{λ} (I_{(i, n)}^{A}) = \sum_{l \geq n} F_{i}^{(l)} V_{A} (λ),$ with $π_{λ} (P) = P v_{λ}$ and $V_{A} (λ) = U_{A}^{-} v_{λ}$ as in The integral residue lattices $L_{A} (\infty)$ and $L_{A} (λ)$ . The latter is an $A$ -submodule inside the field string-image subspace $F_{i}^{n} V (λ)$ , not a $U_{q}$ -submodule or a $U_{A}$ -stable lattice in general; its equality with $V_{A} (λ) \cap F_{i}^{n} V (λ)$ is proved after the adapted global basis construction, not presumed here.

(iii) The integral residue of $I_{(i, n)}^{A} \cap L (\infty)$ is the free $Z$ -span of $B (\infty) \cap {\tilde{f}}_{i}^{n} B (\infty)$ , and the corresponding rational residue of $F_{i}^{n} U_{q}^{-} \cap L (\infty)$ is its $Q$ -span. Integral balancedness of the zero/infinity triple of $I_{(i, n)}^{A}$ is a conclusion of the subsequent string-ideal induction, not an input.

(iv) On every $U_{A}$ -stable integrable module lattice $M_{A}$ , if a label $μ$ satisfies $⟨ h_{i}, μ ⟩ = - n \leq - 1$ , then every $u \in (M_{A})_{μ}$ satisfies the finite identity $u = \sum_{k \geq n} (- 1)^{k - n} {(\binom{k - 1}{k - n})}_{i} F_{i}^{(k)} E_{i}^{(k)} u,$ and consequently $(M_{A})_{μ} = \sum_{k \geq n} F_{i}^{(k)} (M_{A})_{μ + k α_{i}}$ . This is the exact boundary generation supplier used in the string-ideal induction.

Facts & Assumptions

Given: The negative half with skew derivations $e_{i}^{'}, e_{i}^{''}$ , the divided-power form $U_{A}^{-}$ , the crystal lattice $L (\infty)$ with basis $B (\infty)$ , and the integral residue lattice of The integral residue lattices $L_{A} (\infty)$ and $L_{A} (λ)$ .

[F1]

Integral string coefficients (Integral skew-string coefficients on the divided-power negative half): the string decomposition coefficients of $u \in U_{A}^{-}$ are integral: $u_{l} \in U_{A}^{-}$ , $P_{i} (U_{A}^{-}) \subseteq U_{A}^{-}$ , and both $e_{i}^{'}$ and its conjugate preserve $U_{A}^{-}$ .

[F2]

Unique string decomposition (The $i$ -string decomposition of the negative half): every $u \in U_{q}^{-}$ has a unique finite decomposition $u = \sum_{l} F_{i}^{(l)} u_{l}$ with $u_{l} \in \ker e_{i}^{'}$ ; for $u \in F_{i}^{n} U_{q}^{-}$ one has $u_{l} = 0$ for $l < n$ , and conversely.

[F3]

Crystal and projection compatibility (The crystal basis $B (\infty)$ of the negative half, Projection from $L (\infty)$ to $L (λ)$ and its compatibility with the Kashiwara operators): ${\tilde{f}}_{i} L (\infty) \subseteq L (\infty)$ , ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ act on $B (\infty)$ as the crystal operators, and the highest-weight projection is compatible with the crystal operators modulo $q$ .

[F4]

Integral residues (Integral residues of the negative half: freeness, the crystal basis and the form characterization): $L_{Z} (\infty) / q L_{Z} (\infty) = ⨁_{b \in B (\infty)} Z b$ , the residue pairing is $Z$ -valued, and the norm- $1$ elements are signed basis classes.

[F5]

Gaussian calculus (The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality): the Gaussian binomials satisfy the Pascal recurrence and the finite $q$ -binomial theorem used below; all coefficients in the boundary identity are in $A$ .

[F6]

Definitions (Kashiwara's skew derivations $e_{i}^{'}$ and $e_{i}^{''}$ on the negative half, The Lusztig divided-power integral form $U_{A}$ , The Kashiwara lattices $L (λ)$ and $L (\infty)$ and their generating sets): $e_{i}^{'}$ removes one letter, lowers positive height by one and is locally nilpotent; $U_{A}^{-}$ is the $A$ -span of finite products of divided powers; $L (\infty)$ is the $A_{0}$ -lattice generated by the lowering words.

Proof

technique · direct

1.1F1F2algebra

The integral string decomposition and projector. For $u \in U_{A}^{-}$ the unique decomposition of [F2] has $u_{l} = q_{i}^{l (l - 1) / 2} P_{i} (e_{i}^{'})^{l} u$ with $P_{i} = \sum_{l} (- 1)^{l} q_{i}^{- l (l - 1) / 2} F_{i}^{(l)} (e_{i}^{'})^{l}$ , the powers of $e_{i}^{'}$ being ordinary; by [F1] the coefficients $q_{i}^{\pm l (l - 1) / 2}$ and the operators $F_{i}^{(l)}$ and $(e_{i}^{'})^{l}$ preserve $U_{A}^{-}$ , so every $u_{l}$ lies in $U_{A}^{-}$ .

1.2F5algebra

The finite Gaussian identity. Fix integers $m \geq n \geq 1$ , put $z = q_{i}$ , $Q = z^{2}$ , define $(Q; Q)_{r} : = \prod_{j = 1}^{r} (1 - Q^{j})$ for $r \geq 0$ , and set $H (t) = \prod_{j = 1}^{n - 1} (1 - Q^{- j} t) \prod_{j = 1}^{m - n} (1 - Q^{j} t)$ , a polynomial of degree $m - 1$ . For every polynomial of degree $\leq m - 1$ the finite $q$ -binomial theorem of [F5] gives $\sum_{k = 0}^{m} (- 1)^{k} Q^{k (k - 1) / 2 - k (m - 1)} {(\binom{m}{k})}_{Q} H (Q^{k}) = 0$ : on the monomial $t^{s}$ the sum factors as $\prod_{j = 0}^{m - 1} (1 - Q^{j + s - m + 1})$ , which contains the zero factor $j = m - 1 - s$ . For $1 \leq k < n$ one has $H (Q^{k}) = 0$ ; for $k \geq n$ , $H (Q^{k}) / [(Q; Q)_{n - 1} (Q; Q)_{m - n}] = {(\binom{k - 1}{n - 1})}_{Q} {(\binom{m - n + k}{k})}_{Q}$ ; and the $k = 0$ term equals $(- 1)^{n - 1} Q^{- n (n - 1) / 2}$ . Multiplying the vanishing sum by $(- 1)^{n} Q^{n (n - 1) / 2}$ divided by the two factorials turns it into $\sum_{k = n}^{m} (- 1)^{k - n} {(\binom{k - 1}{k - n})}_{z} {(\binom{m - n + k}{k})}_{z} {(\binom{m}{k})}_{z} = 1$ , using ${(\binom{a}{b})}_{z} = z^{- b (a - b)} {(\binom{a}{b})}_{Q}$ and the exponent identity $k (k - 1) / 2 - k (m - 1) + n (n - 1) / 2 = - \frac{(n - 1) (k - n) + k (m - n) + k (m - k)}{2}$ .

2.1F2F6step 1.1

String ideals. By [F2], $u \in F_{i}^{n} U_{q}^{-}$ if and only if $u_{l} = 0$ for $l < n$ ; hence $I_{(i, n)} = U_{A}^{-} \cap F_{i}^{n} U_{q}^{-}$ equals $⨁_{l \geq n} F_{i}^{(l)} (\ker e_{i}^{'} \cap U_{A}^{-})$ directly, and this equals the sum $\sum_{l \geq n} F_{i}^{(l)} U_{A}^{-}$ because each displayed direct summand is contained in $F_{i}^{(l)} U_{A}^{-}$ , while multiplying an integral string expansion by $F_{i}^{(l)}$ yields only positions $l + m \geq l$ with integral coefficients ${(\binom{l + m}{l})}_{i}$ ; the sum is not direct as written. Applying the $A$ -linear projection $π_{λ}$ of The integral residue lattices $L_{A} (\infty)$ and $L_{A} (λ)$ gives $J_{(i, n)}^{A} = \sum_{l \geq n} F_{i}^{(l)} V_{A} (λ)$ , an $A$ -submodule of the field subspace $F_{i}^{n} V (λ)$ , with no $U_{q}$ - or $U_{A}$ -stability asserted.

2.2F5F6step 1.2

Boundary generation on a module lattice. By Uniqueness of the $i$ -string decomposition of an integrable module, reduce to an admissible string vector $u = F_{i}^{(m)} v$ with $E_{i} v = 0$ and top weight $2 m - n$ . Admissibility gives $m \leq 2 m - n$ , hence $m \geq n$ ; the formulas of Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules give $F_{i}^{(k)} E_{i}^{(k)} u = {(\binom{m - n + k}{k})}_{z} {(\binom{m}{k})}_{z} u$ for $k \leq m$ and $0$ for $k > m$ . Substituting step 1.2, the alternating sum of step (iv) is $u$ : the coefficients are exactly those of the proved identity, with finite support because $k \leq m$ . Every $u \in (M_{A})_{μ}$ is a finite sum of such string vectors, and all operators and coefficients involved lie in $A$ and preserve $M_{A}$ (The Lusztig divided-power integral form $U_{A}$ ), so $(M_{A})_{μ} = \sum_{k \geq n} F_{i}^{(k)} (M_{A})_{μ + k α_{i}}$ , with finiteness supplied by integrability.

3.1F1F2F3F4step 2.1

String-ideal residues. By [F2], the field image of ${\tilde{f}}_{i}^{n}$ is $F_{i}^{n} U_{q}^{-}$ . On this image ${\tilde{f}}_{i}^{n} {\tilde{e}}_{i}^{n}$ is the identity. Both shifts preserve $U_{A}^{-}$ and $L (\infty)$ by [F1], [F3], so $I_{(i, n)}^{A} \cap L (\infty) = {\tilde{f}}_{i}^{n} L_{Z}$ . Taking residues and using [F4] gives exactly the integral span of ${\tilde{f}}_{i}^{n} B (\infty)$ ; the same argument over $A_{0}$ gives its rational span.

4.1step 1.1step 2.1step 3.1step 2.2∎

Conclusion. Steps 1.1–2.1 prove (i) and (ii), step 3.1 proves (iii), step 2.2 proves (iv); integral balancedness of the string-ideal triple is left to the subsequent induction and no saturation of $J_{(i, n)}^{A}$ , no $U_{q}$ -structure on it and no divided-derivation integrality are asserted.

Remarks

The ordinary powers of $e_{i}^{'}$ in the projector are essential: in rank one the divided expression $(e_{i}^{'})^{2} / [2]_{i}$ applied to $F_{i}^{(2)}$ gives $1 / (1 + q_{i}^{2})$ , which is not in $A$ ; the integrality of the decomposition rests instead on the ordinary-power skew-string supplier [F1] and the exact finite Gaussian identity of step 1.2.

Depends on

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Sources

Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Mathematical Journal 63 (1991), 465–516 (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)
George Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110, Birkhauser, 1993; second printing 1994; author-hosted scan) (standard reference, not scraped)