Why didn't the writer mention anything about Emily's mother?
We don't really know why the author doesn't mention Emily's mother. However, from a literary standpoint, the omission of any possible maternal figure clearly illustrates the stark reality of Emily's marginalized existence.
Emily's father appears to be a pervasive, controlling influence in her life. The image of a fragile Emily, juxtaposed against a virile figure of paternal dominance, is telling. It clearly demonstrates the power Emily's father exerted over her.
We had long thought of them as a tableau, Miss Emily a slender figure in white in the background, her father a spraddled silhouette in the foreground, his back to her and clutching a horsewhip, the two of them framed by the back-flung front door.
Emily's life choices have been so dictated by her father that she finds herself at a loss after his death. Having been denied any significant autonomy throughout her life, Emily finds herself infatuated with another inaccessible male figure after her father dies.
When she had first begun to be seen with Homer Barron, we had said, "She will marry him." Then we said, "She will persuade him yet," because Homer himself had remarked--he liked men, and it was known that he drank with the younger men in the Elks' Club--that he was not a marrying man.
With no mother in the picture, Emily's descent to a sort of cathartic madness becomes more starkly evident. Her dysfunctional view of love is an emblem of her father's heritage; there is no recourse to stability in the form of a mitigating, maternal influence. Faced with the possibility of losing Homer, Emily poisons him and keeps his corpse in her home until her own death. For once in her life, Emily has retained a sort of elusive control; however its quality is marred by its semblance to the flawed judgments of her father's soul-crushing paternalism. Emily may have kept Homer by her side, but she never gets to experience a satisfying relationship with her poisoned lover.
Hence, the author's omission of a maternal figure or mother in the story serves as a literary device to illuminate the protagonist's suffering and descent to madness in the face of dysfunction and oppression.