Geodesic
A new way to construct $1$-singular Gelfand-Tsetlin modules
Algebra and discrete mathematics, Tome 23 (2017) no. 1, pp. 180-193

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We present a simplified way to construct the Gelfand-Tsetlin modules over $\mathfrak{gl}(n,\mathbb{C})$ related to a $1$-singular GT-tableau defined in [6]. We begin by reframing the classical construction of generic Gelfand-Tsetlin modules found in [3], showing that they form a flat family over generic points of $\mathbb{C}^{\binom{n}{2}}$. We then show that this family can be extended to a flat family over a variety including generic points and $1$-singular points for a fixed singular pair of entries. The $1$-singular modules are precisely the fibers over these points.
Keywords: Gelfand-Tsetlin modules, Gelfand-Tsetlin bases, tableaux realization. 
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     author = {Pablo Zadunaisky},
     title = {A new way to construct $1$-singular {Gelfand-Tsetlin} modules},
     journal = {Algebra and discrete mathematics},
     pages = {180--193},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2017_23_1_a8/}
}
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Pablo Zadunaisky. A new way to construct $1$-singular Gelfand-Tsetlin modules. Algebra and discrete mathematics, Tome 23 (2017) no. 1, pp. 180-193. http://geodesic.mathdoc.fr/item/ADM_2017_23_1_a8/