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The four Fock bases in a small two-factor tensor space

Example

Take $I = {0}$ , $I^{+} = {0, 1}$ and $Λ = 2 Λ_{0}$ in the conventions of The higher-level wedge-tensor Fock module, so that $F = V \otimes V$ has the four monomials $M_{00}, M_{01}, M_{10}, M_{11}$ in the source normalization of The dual-canonical, canonical and quasi-canonical Fock bases.

In the mixed order $(M_{01}, M_{10})$ , the ordinary bar matrix is $B_{ψ} = (\begin{array}{cc} 1 & s^{- 1} - s \\ 0 & 1 \end{array}),$ and both source monomial Gram matrices equal $S = B_{ψ}^{- 1}$ , i.e. $S = (\begin{array}{cc} 1 & s - s^{- 1} \\ 0 & 1 \end{array})$ . All unchanged labels satisfy $L = T = M$ . The remaining vectors are $L_{10} = M_{10} - s M_{01}, T_{10} = M_{10} + s^{- 1} M_{01}, P_{01} = M_{01} + s M_{10}, P_{10} = M_{10},$ and the reverse family has ${\tilde{L}}_{01} = {\tilde{M}}_{01} - s {\tilde{M}}_{10}$ .

The transition and pairing matrices satisfy ${\overline{P}}^{t} S L = I, {\tilde{L}}^{t} S T = I$ on the four-dimensional space, the four whole families are pairwise different over $Q (s)$ , and both $L$ and $T$ are ordinary-bar fixed, unlike the general quasi-canonical family.

At $t = s^{- 1}$ the highest-module form rescaling is $B_{8} (T_{10}, T_{10}) = 1 + t^{2}, D_{01} = \frac{T_{10}}{s + s^{- 1}}, G_{u p} = s D_{01} .$

Facts & Assumptions

Given: The rank-one Fock model $I^{+} = {0, 1}$ , $Λ = 2 Λ_{0}$ , its four monomials in the mixed order $(M_{01}, M_{10})$ , the source bar $ψ$ and families $L, T, P$ of The dual-canonical, canonical and quasi-canonical Fock bases, the forms $⟨ -, - ⟩$ , $C$ , $C_{0}$ and $B_{8}$ , and the projection $π$ of Standard tableau monomials form a basis of the highest weight Fock submodule.

[F1]

Source bar and families (The dual-canonical, canonical and quasi-canonical Fock bases, The higher-level wedge-tensor Fock module): the monomial basis and its weights; the triangular charts $d, p$ ; the delta pairings $⟨ P_{A}, L_{B} ⟩ = δ_{A B}$ , $C ({\tilde{L}}_{A}, T_{B}) = δ_{A B}$ .

[F2]

The normalized quasi- $R$ supplier Existence and uniqueness of the weightwise quasi- $R$ intertwiner gives $Θ_{α_{0}} = (t - t^{- 1}) E_{0} \otimes F_{0}$ ; the tensor bar is supplied by The tensor-product bar on the Fock module and its truncation compatibility, with $t = s^{- 1}$ . Its fixed-vector equation is $c = B_{ψ} \overset{ˉ}{c}$ .

[F3]

Highest constituent and rescaling (Standard tableau monomials form a basis of the highest weight Fock submodule, The highest weight Fock submodule carries the quantum-group global bases): the projection $π$ , the source dual-canonical basis on ${S t d}^{Λ}$ , and the weight rescaling $B_{8} = s^{- a} C_{0}$ with $G_{u p} (A) = s^{a} D_{A}$ .

Verification

technique · direct

1.1F2algebra

The bar matrix. Each natural-factor bar fixes its two coordinate vectors. The only nonconstant quasi- $R$ term acting on this tensor square is $(s^{- 1} - s) E_{0} \otimes F_{0}$ ; it takes $M_{10}$ to $(s^{- 1} - s) M_{01}$ and kills the other three monomials. Higher terms vanish because $E_{0}^{2} = F_{0}^{2} = 0$ on a natural factor. This gives the displayed $B_{ψ}$ in the mixed order $(M_{01}, M_{10})$ .

2.1F1step 1.1algebra

The four families. A vector $M_{10} + c M_{01}$ is bar fixed exactly when $c - \overset{ˉ}{c} = s^{- 1} - s$ . The two triangular charts give $c = - s$ for the $s Z [s]$ -chart and $c = s^{- 1}$ for the $s^{- 1} Z [s^{- 1}]$ -chart, that is $L_{10} = M_{10} - s M_{01}$ and $T_{10} = M_{10} + s^{- 1} M_{01}$ . The quasi-canonical family is defined by the inverse transition matrix, giving $P_{01} = M_{01} + s M_{10}$ and $P_{10} = M_{10}$ ; the reverse reading gives ${\tilde{L}}_{01} = {\tilde{M}}_{01} - s {\tilde{M}}_{10}$ . Since the off-diagonal entries are $- s$ , $s^{- 1}$ and $s$ , the four families are pairwise different over $Q (s)$ .

3.1F1step 2.1algebra

The Gram matrices and delta identities. The defining forms have labelled monomial Gram matrices equal to the inverse ordinary-bar matrix, so $S = B_{ψ}^{- 1} = (\begin{array}{cc} 1 & s - s^{- 1} \\ 0 & 1 \end{array})$ for both forms, as displayed. Substituting the coefficient matrices of step 2.1 and multiplying the resulting $4 \times 4$ matrices (which are the identity on the $00$ - and $11$ -labelled directions) gives ${\overline{P}}^{t} S L = I$ and ${\tilde{L}}^{t} S T = I$ exactly; these are the delta identities $⟨ P_{A}, L_{B} ⟩ = δ_{A B}$ and $C ({\tilde{L}}_{A}, T_{B}) = δ_{A B}$ of [F1].

3.2F3step 2.1algebra

The form rescaling. The mixed-weight vector in the highest constituent is $T_{10} = F_{0} M_{00}$ ; its norm under the normalized bilinear form $C_{0}$ is $[2]_{s} = s + s^{- 1}$ , and by [F3] the library polarization gives $B_{8} (T_{10}, T_{10}) = s^{- 1} [2]_{s} = 1 + t^{2}$ at $t = s^{- 1}$ . The projected dual-canonical vector is $D_{01} = π L_{01} = T_{10} / (s + s^{- 1})$ , and the library form-dual upper vector is its rescaled $G_{u p} = s D_{01}$ , which checks the exact rescaling of [F3] without asserting that the source $D$ -family itself is the library upper family.

4.1step 1.1step 2.1step 3.1step 3.2∎

Conclusion. In this four-dimensional model the ordinary bar matrix is $B_{ψ}$ , both Gram matrices are $S = B_{ψ}^{- 1}$ , the displayed $L, T, P, \tilde{L}$ vectors are the four distinct source families with the stated delta pairings, and the highest-component rescaling checks $B_{8} (T_{10}, T_{10}) = 1 + t^{2}$ , $D_{01} = T_{10} / (s + s^{- 1})$ and $G_{u p} = s D_{01}$ .

Depends on

Used by

The canonical and quasi-canonical bases need not coincide Counterexample

Dependency tree · two levels

64 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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)
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)