Let $\mathbf{k}$ be an algebraically closed field of prime characteristic and $S(n)$ be the special Lie superalgebra of Cartan type over $\mathbf{k}$. Define $\bar{S}(n)=S(n)\oplus\mathbf{k}\mbox{-}\{\xi_1D_1 \}$. So $\bar{S}(n)_0\cong\mathfrak{gl}(n)$. Let $\ggg=S(n)$ or $\bar{S}(n)$. We investigate in this paper the representations of $\ggg$ when $\chi$ is restricted or $\mathrm{ht}(\chi)=1$. The main results are listed below.
(1) When $\mathrm{ht}(\chi)=1$, the irreducible representations of $U_{\chi}(\ggg)$ are considered. Precisely, the composition factors of the Kac modules are confirmed and the dimensions of simple modules are given.
(2) When $\chi=0$ or $\mathrm{ht}(\chi)=1$, the structures of indecomposable projective modules are studied and the Cartan invariants of $U_{\chi}(\frak g)$ are given.
(3) When $\chi=0$ or $\mathrm{ht}(\chi)=1$, we show that the representation category over $U_{\chi}(\ggg)$ has only one block (reckoning parities in).
@article{JOLT_2023_33_3_a10,
author = {F. Duan},
title = {Representations of the {Special} {Lie} {Superalgebra} with {p-Character} of {Height} {One
}},
journal = {Journal of Lie Theory},
pages = {887--918},
year = {2023},
volume = {33},
number = {3},
doi = {10.5802/jolt.1311},
zbl = {1536.17011},
language = {en},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1311/}
}
TY - JOUR AU - F. Duan TI - Representations of the Special Lie Superalgebra with p-Character of Height One JO - Journal of Lie Theory PY - 2023 SP - 887 EP - 918 VL - 33 IS - 3 UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.1311/ DO - 10.5802/jolt.1311 LA - en ID - JOLT_2023_33_3_a10 ER -
F. Duan. Representations of the Special Lie Superalgebra with p-Character of Height One. Journal of Lie Theory, Volume 33 (2023) no. 3, pp. 887-918. doi: 10.5802/jolt.1311
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