  The Kronecker coefficient g_{\lambda \mu \nu} is the multiplicity of the
GL(V)\times GL(W)-irreducible V_\lambda \otimes W_\mu in the restriction of the
GL(X)-irreducible X_\nu via the natural map GL(V)\times GL(W) \to GL(V \otimes
W), where V, W are \mathbb{C}-vector spaces and X = V \otimes W. A fundamental
open problem in algebraic combinatorics is to find a positive combinatorial
formula for these coefficients.
  We construct two quantum objects for this problem, which we call the
nonstandard quantum group and nonstandard Hecke algebra. We show that the
nonstandard quantum group has a compact real form and its representations are
completely reducible, that the nonstandard Hecke algebra is semisimple, and
that they satisfy an analog of quantum Schur-Weyl duality.
  Using these nonstandard objects as a guide, we follow the approach of Adsul,
Sohoni, and Subrahmanyam to construct, in the case dim(V) = dim(W) =2, a
representation \check{X}_\nu of the nonstandard quantum group that specializes
to Res_{GL(V) \times GL(W)} X_\nu at q=1. We then define a global crystal basis
+HNSTC(\nu) of \check{X}_\nu that solves the two-row Kronecker problem: the
number of highest weight elements of +HNSTC(\nu) of weight (\lambda,\mu) is the
Kronecker coefficient g_{\lambda \mu \nu}. We go on to develop the beginnings
of a graphical calculus for this basis, along the lines of the U_q(\sl_2)
graphical calculus, and use this to organize the crystal components of
+HNSTC(\nu) into eight families. This yields a fairly simple, explicit and
positive formula for two-row Kronecker coefficients, generalizing a formula of
Brown, van Willigenburg, and Zabrocki. As a byproduct of the approach, we also
obtain a rule for the decomposition of Res_{GL_2 \times GL_2 \rtimes \S_2}
X_\nu into irreducibles.
