Sasha straight up tells us readers that this scarf is a Big Symbol. The "three tips" of the scarf stand for the three generations: "mature Communists, the Communist Youth, and the Young Pioneers" (4.11). Its color (red) stands for "blood spilled for the cause of the Communist Party" (4.14). Symbol decoded, right?
Ah, but there's more to it than that.
What about the reflection of the scarf in Sasha's dad's glasses? Here's what Sasha sees: "I see myself reflected in his glasses; scarlet burns at my throat" (4.18). It's a proud moment for Sasha, but that word "burns" sure throws an ominous note into the mix. A bit of foreshadowing, perhaps?
Sasha's dad has just tied the scarf around his neck, and now he sees it reflected in his glasses, which draws a parallel between the scarf and his dad. Which makes sense, since the scarf basically embodies the ideals of the type of person Sasha wants to be: "My dad is a hero and a Communist and, more than anything, I want to be like him" (1.1). His dad's also going to be the one to officially tie the scarf around his neck and make him a Pioneer the next day. Or at least that's the plan.
Initially, what the scarf symbolizes is positive to Sasha, and it's super important to him because of that. In fact, it's the only item he takes from his apartment when he's booted out (17.2). But in the end, after all of the lessons Sasha has learned (check out the "Character" section for those tidbits), he rejects what the scarf stands for, and declines to become a member of the Young Soviet Pioneers.
We never find out for sure what happens to this once-beloved scarf, though. The last thing we hear about it is when Sasha notes that it's tucked away in his pocket, and that it was the only thing he took from the apartment when he was kicked out (17.2). We think it's safe to assume that it's probably still tucked away in his pocket, forgotten, like his old dream.
But here's some food for thought: in the end, the red Pioneer scarf is symbolically replaced by the "woolen scarf" the nice lady gives him while they wait in line at Lubyanka prison (30.10). This tiny moment represents a new future and a new home for Sasha, where scarves are scarves, instead of political symbols.