The exact nature of Joseph's special tunic (Gen 37:3) may never be decided. Of course, many scholars now go with "a long robe with sleeves" (see NRSV, REB, NAB), understanding the Hebrew term פס in a literal sense of "flat of hand or foot," so that the tunic would be one reaching palms and soles. I myself am skeptical. Prof. Chris Heard in a post
some time ago remarked that this reading would make Joseph something of an objectionable figure, who was in a special category exempt from hard work. Chris wrote, "Joseph’s כתנת הפסים, then, marks—in its context in Genesis 37—a kind of exemption from the labor to which his brothers were otherwise subject." To me, Joseph is less of a spoiled brat in the story than a young, naive, and unpopular figure, who shows himself completely faithful to his father, Jacob.
The appearance of the tunic on the central (bending) figure in the upper panel of the image above suggests one of the other major translation options for Gen 37:3, i.e., a tunic with varied pieces, panels, or bands (other possible senses of פס) of (multicolored?) fabric (cf. the LXX rendering). The well-known Egyptian wall painting is from the tomb of Khnumhotep III at Beni Hasan, a 19th century image. The figure with the special tunic is the leader of the caravan of hyksos/asiatics bringing eye-salve into Egpyt for trade, a ruler named Ab-Sha. What strikes me about Ab-Sha and his tunic is that the brightly colored garment fit for a leader here is not understood as anything incompatible with labor, at least the labor of directing animals in a trade caravan. As with the other caravan members, some of whom have no top on at all, his arms appear free to move and work.
The kind of special tunic of Joseph is worn by one other figure in Bible, the king's daughter Tamar in 2 Samuel 13. As R. E. Freidman has argued, the two tunics (Joseph's and Tamar's) may well form parts of one story, one large interconnected source/narrative. The tunics may symbolize the same thing in both Genesis and 2 Samuel. Freidman writes, "It is hardly coincidental that the two people who wear a coat of many colors in the Bible are both victims of violence by their brothers, and that both coats are torn. The significance of the coat, therefore, is as a symbol of injustice among siblings." It appears to me that a theme of this large source/narrative is God's oracle/dream/providence sustaining itself despite human/sibling rivalry, jealousy, and violence. The theme is better supported if the coat signifies victimization rather than someone living a spoiled life or actually putting on airs.