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The canonical and quasi-canonical bases need not coincide

Statement refuted

In the four-dimensional model of The four Fock bases in a small two-factor tensor space the canonical and quasi-canonical Fock families coincide on every label.

Facts & Assumptions

Given: The rank-one Fock model with monomials $M_{00}, M_{01}, M_{10}, M_{11}$ , the source families $L, T, P$ of The dual-canonical, canonical and quasi-canonical Fock bases, and the computed bar matrix and vectors of The four Fock bases in a small two-factor tensor space.

[F1]

Bar and families (The tensor-product bar on the Fock module and its truncation compatibility, The dual-canonical, canonical and quasi-canonical Fock bases): $ψ$ is triangular with $ψ M_{10} = M_{10} + (s^{- 1} - s) M_{01}$ and fixes the other monomials; the canonical family is the $s^{- 1} Z [s^{- 1}]$ -chart and the quasi-canonical family is defined by the inverse transition matrix.

[F2]

Highest constituent (The highest weight Fock submodule carries the quantum-group global bases, Standard tableau monomials form a basis of the highest weight Fock submodule, The four Fock bases in a small two-factor tensor space): the projection $π$ , the rectified highest label $01^{↑} = 10$ , and the rescaling $P_{A} = s^{a} T_{A^{↑}}$ on the highest constituent with $a = 1$ .

Counterexample

Take $I = {0}$ , $I^{+} = {0, 1}$ , $Λ = 2 Λ_{0}$ , and retain the mixed order $(M_{01}, M_{10})$ and the notation of The dual-canonical, canonical and quasi-canonical Fock bases and The four Fock bases in a small two-factor tensor space. There $P_{01} = M_{01} + s M_{10} = s T_{10},$ because $T_{10} = M_{10} + s^{- 1} M_{01}$ ; but $P_{10} = M_{10} \neq M_{10} + s^{- 1} M_{01} = T_{10}$ over $Q (s)$ . Hence the canonical and quasi-canonical families are different, and so are the dual-canonical and monomial families. The highest-component label rectification is $01^{↑} = 10$ with $a = 1$ , which exhibits the genuine power and label distinction: the coincidence $P_{01} = s T_{10}$ happens on the rectified highest label, not by an identity of the two index sets. At $s = 1$ the highest-submodule vectors $P_{01}$ and $T_{10}$ agree after rectification, but the full canonical and quasi-canonical families are still not interchangeable. This is no claim that every corresponding label differs at $s = 1$ .

Proof technique: direct.

1.1F1givenalgebra

The two families at label $10$ . In the mixed order the bar fixes $M_{01}$ and sends $M_{10}$ to $M_{10} + (s^{- 1} - s) M_{01}$ . A vector $M_{10} + c M_{01}$ is fixed exactly when $c - \overset{ˉ}{c} = s^{- 1} - s$ . The positive chart gives $c = - s$ and the negative chart gives $c = s^{- 1}$ ; uniqueness follows because a bar-fixed polynomial in either $s Z [s]$ or $s^{- 1} Z [s^{- 1}]$ is zero. Thus $L_{01} = T_{01} = M_{01}$ , $L_{10} = M_{10} - s M_{01}$ and $T_{10} = M_{10} + s^{- 1} M_{01}$ . Inverting the $L$ chart gives $M_{10} = L_{10} + s L_{01}$ , so the transposed inverse chart is $P_{01} = M_{01} + s M_{10}$ and $P_{10} = M_{10}$ . Linear independence of the two monomials proves $P_{10} \neq T_{10}$ .

2.1F1F2step 1.1algebra

The rectified highest-label coincidence. At label $01$ the same computations give $P_{01} = M_{01} + s M_{10} = s T_{10}$ , so the two families do meet at the rectified highest label with the power $a = 1$ of [F2]; this is an identity between specific vectors, not an identification of the two labelled families.

3.1step 1.1step 2.1∎

Conclusion. Since $P_{10} \neq T_{10}$ , the canonical and quasi-canonical families are different over $Q (s)$ ; the dual-canonical and monomial families differ too, because $L_{10} = M_{10} - s M_{01} \neq M_{10}$ . The highest-submodule coincidence $P_{01} = s T_{10}$ is exactly the rectified normalization of [F2], and at $s = 1$ it does not make the whole families interchangeable.

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Sources

Jonathan Brundan and Alexander Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z. 266 (2010) 877-919 (complete 44-page arXiv article) (standard reference, not scraped)
Jonathan Brundan, Dual canonical bases and Kazhdan-Lusztig polynomials, J. Algebra 306 (2006) 17-46 (complete 29-page arXiv v2 article) (standard reference, not scraped)