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Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem
Computational Complexity · Submitted 22 Mar 2007
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The authors' main claim This yields a fairly simple, explicit and positive formula for two-row Kronecker coefficients, generalizing a formula of Brown, van Willigenburg, and Zabrocki.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.06 | 0.02 | asserted 1.00 |
| s1 | A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. | gap 0.69 | 0.03 | 0.02 | asserted 1.00 |
| s2 | We construct two quantum objects for this problem, which we call the nonstandard quantum group and nonstandard Hecke algebra. | contribution 1.00 | 0.96 | 0.07 | asserted 1.00 |
| s3 | 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. | result 0.90 | 0.95 | 0.11 | asserted 1.00 |
| s4 | 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. | method 0.74 | 0.96 | 0.06 | asserted 1.00 |
| s5 | 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}. | result 0.85 | 0.96 | 0.66 | asserted 1.00 |
| s6 | 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. | method 0.51 | 0.95 | 0.04 | asserted 1.00 |
| s7 | This yields a fairly simple, explicit and positive formula for two-row Kronecker coefficients, generalizing a formula of Brown, van Willigenburg, and Zabrocki. | result 0.97 | 0.94 | 0.51 | asserted 1.00 |
| s8 | 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. | result 0.96 | 0.94 | 0.05 | asserted 1.00 |
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Original abstract
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.
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4 arXiv papers cite this in 20 sentences
Papers citing it · 11 sentences without a confident label
Compares with it 1 paper
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The construction in this paper has many similarities with that in [ref] .
Quantum deformations of the restriction of $GL_{mn}(\C)$-modules to $GL_m(\C) \times GL_n(\C)$ · Introduction · 1.00
Cites it as background 3 papers
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Similarly, positive combinatorial interpretations of the Kronecker coefficients $g(ł,,)$ exist only in particular cases: (i) if $ł$ and $$ are both hooks [ref] ; (ii) if $ł$ is a two row partition (with some conditions on the size of the first part) [ref] ; (iii) if $λ$ is a hook partition [ref] ; (iv) if $ł$ and $$ are both two row partition [ref] .
Stability of coefficients in the Kronecker product of a hook and a rectangle · Introduction · 0.88
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We recall the description of the $S^2_̋r$-irreducibles from [ref] .
Representation theory of the nonstandard Hecke algebra · Representation theory of $S^2_̋r$ · 0.84
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For some known special cases, see [ref] .
A simplified Kronecker rule for one hook shape · Introduction · 0.99
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