Alignment conjecture for virtual crystals Vr,sV^{r,s}

For 4g44\mathfrak{g}4 of type Dn+1(2)D_{n+1}^{(2)}, A2n(2)A_{2n}^{(2)}, A2n(2)A_{2n}^{(2)\dagger}, or Cn(1)C_n^{(1)}, let Vr,sV^{r,s} be the virtual crystal embedded in an ambient type A2n1(1)A_{2n-1}^{(1)} crystal. An aligned virtual crystal is one whose inherited intrinsic energy agrees with the prescribed virtual grading under the alignment conditions defined in the paper. Alignment conjecture. Vr,sV^{r,s} is aligned. Alignment is needed for the virtual crystal to inherit a compatible intrinsic energy and supports the subsequent identification with the desired affine crystal; the source leaves this property conjectural.

Sources & referencesView supporting material

Primary source

Masato Okado, Anne Schilling and Mark Shimozono, “Virtual crystals and fermionic formulas of type D_n+1^(2), A_2n^(2), and C_n^(1)”, arXiv:math/0105017 (2001).

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