## Artin’s theorem on induced representations and the kernel

Let $$G$$ be a finite group, and let $$X$$ be a family of subgroups of $$G$$ closed under conjugation and under passage to subgroups. Suppose further that $$G$$ is the union of the elements of $$X,$$ and denote by $$R(G)$$ the representation ring. By Artin’s theorem, we have a surjective map $$Ind: \oplus_{H \in X} \mathbb{Q} \otimes R(H) \rightarrow \mathbb{Q} \otimes R(G),$$ given by taking the induced character and extending by linearity. I will now define two special classes of elements in the source of this map.

Elements of Class 1: Let $$H,H’ \in X, H’ \subset H,$$ and let $$\psi’ \in R(H’).$$ Let $$\psi = Ind_{H’}^H \psi’.$$ Then elements of class $$1$$ are of the form $$\psi-\psi’.$$

Elements of class 2: Let $$H \in X, s \in G$$ and set $$H^s = sHs^{-1}.$$ Let $$\psi \in R(H),$$ and define $$\psi^s \in R(H^s)$$ by $$\psi^s(shs^{-1})= \psi(h).$$ Then elements of class $$2$$ are of the form $$\psi-\psi’.$$

It is easy to see that all elements of class $$1$$ and $$2$$ lie in the kernel of the map $$Ind.$$ I want to prove that in fact, the kernel is generated by elements of class $$1$$ and $$2.$$ The hint I have seem is that one should base change go $$\mathbb{C},$$ and then use duality. It is then claimed that one is reduced to proving that if for every $$H \in X$$ one has a class function $$f_H$$ satisfying restriction snd conjugation conditions as above, then there exists a class function $$f$$ on $$G$$ restricting to $$f_H$$ for all $$X.$$.

I supppse we should use Frobenius reciprocity in some clever way, but I do not see how the statement reduces to what I just claimed, so I would appreciate help, or full solutions.

## Artin’s theorem on induced representation snd the kernel

Let $$G$$ be a finite group, and let $$X$$ be a family of subgroups of $$G$$ closed under conjugation and under passage to subgroups. Suppose further that $$G$$ is the union of the elements of $$X,$$ and denote by $$R(G)$$ the representation ring. By Artin’s theorem, we have a surjective map $$Ind: \oplus_{H \in X} \mathbb{Q} \otimes R(H) \rightarrow \mathbb{Q} \otimes R(G),$$ given by taking the induced character and extending by linearity. I will now define two special classes of elements in the source of this map.

Elements of Class 1: Let $$H,H’ \in X, H’ \subset H,$$ and let $$\psi’ \in R(H’).$$ Let $$\psi = Ind_{H’}^H \psi’.$$ Then elements of class $$1$$ are of the form $$\psi-\psi’.$$

Elements of class 2: Let $$H \in X, s \in G$$ and set $$H^s = sHs^{-1}.$$ Let $$\psi \in R(H),$$ and define $$\psi^s \in R(H^s)$$ by $$\psi^s(shs^{-1})= \psi(h).$$ Then elements of class $$2$$ are of the form $$\psi-\psi’.$$

It is easy to see that all elements of class $$1$$ and $$2$$ lie in the kernel of the map $$Ind.$$ I want to prove that in fact, the kernel is generated by elements of class $$1$$ and $$2.$$ The hint I have seem is that one should base change go $$\mathbb{C},$$ and then use duality. It is then claimed that one is reduced to proving that if for every $$H \in X$$ one has a class function $$f_H$$ satisfying restriction snd conjugation conditions as above, then there exists a class function $$f$$ on $$G$$ restricting to $$f_H$$ for all $$X.$$.

I supppse we should use Frobenius reciprocity in some clever way, but I do not see how the statement reduces to what I just claimed, so I would appreciate help, or full solutions.

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## Force re-signing of kernel and modules for Secure Boot?

I accidentally hit reboot and the next boot failed signatures in Secure Boot. I disabled secure boot to get back in.

How can I force re-signing of everything so that secure boot can be enabled again?

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## journald flooded by kernel messages

My journald log was at some point filled up with kernel messages like this.

Mar 12 05:03:53 workstation.mydomain kernel: FAT-fs (sdd1): FAT read failed (blocknr 1785)

Each followed by a message like this

Mar 12 05:03:53 workstation.mydomain systemd-journal: Missed 206 kernel messages

This was likely due to connecting some faulty drive. Anyhow, I got several millions of these messages causing the log to fill up within minutes.

The journald.conf settings RateLimitInterval, RateLimitBurst does not seem to apply to kernel messages as the rate widely exceeds the default values for these parameters (which is what I use).

How can I control this? Is there some separate settings for kernel messages?

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## upadete kernel while installing using usb

i have an issue for installing Manjaro for my pc. the stable version using 4.19, and get stuck at black screen while installing (i915 ok … ).

after searching on a forum, that says the i915 error is fixed on 4.20 but i can’t get the Manjaro with 4.20, all of the previews version are 404

so, can we update the kernel in my bootable USB? i was tried, download the kernel 420 (ctrl+alt+f4), but for change kernel we need restart the pc, after restart the 420 kernel are gone ( i thinks coz while installing, the system mount to RAM, maybe #newbie)

some forums say 4.20 available at pure Arch Linux, but i don’t wanna use the pure arch.

i9 9900k, msi Z390, Manjaro gnome 18.04

thanks for help

## Cannot GRUB boot kernel v. 5.0.3 because of invalid signature

I am attempting to boot to a newer kernel because of an audio problem caused by a bug in the kernel discussed here.

I used Ukuu to get and install kernel v. 5.0.3. Upon updating GRUB and attempting to reboot into the new version, GRUB says that it can not boot it because it has an invalid signature.

What is the simplest way to fix this? Can I make it so that I don’t need a signed kernel to boot?

Thanks

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## What is the Virtuallbox guest additions kernel module called?

I installed Virtualbox 6.0 on my laptop which is UEFI with SecureBoot enabled, so I had the added complication of having to sign the kernel modules, which I identified as:

• vboxdrv
• vboxnetflt
• vboxpci

However I overlooked the fact that the guest additions is also a kernel module and has to be signed, seems strange that I can’t seem to find something as simple as the module name, which I am guessing is vboxguest?

so I can sign it with:

sudo /usr/src/linux-headers-$(uname -r)/scripts/sign-file sha256 ./MOK.priv ./MOK.der$  (modinfo -n vboxguest) 

Yes?

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## In a Ubuntu release version, what kinds of updates/upgrades can we (not) get on installed packages and Linux kernel?

In Debian Stable, although I haven’t experienced it, I heard that the versions of installed packages and kernel can be increased. I am not sure in what sense that means. A version number usually consists of three fields. I guess it might mean we can’t update or upgrade an installed package or Linux kernel in the first field, but only in in the last two fields.

In a Ubuntu LTS (e.g. Lubuntu 18.04), what kinds of updates/upgrades can we get on installed packages and Linux kernel? What kinds can’t we get? In this aspect, which one is it more similar to: Debian Stable, Testing or Unstable?

In a Ubuntu nonLTS version, similar questions. In this aspect, which one is it more similar to: Debian Stable, Testing or Unstable?

Thanks.

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## Salix Won’t Boot – Kernel Panic: Not Syncing

I have an old 2006 Macbook Pro A1181 laptop I’ve been trying to get up and running for a while. I’ve tried many options (Fedora, Bunsen Labs, Vector Linux, Porteus, ToriOS and Salix), but none would install properly. After install, the mac would sit there, flashing a folder with a question-mark. Porteus worked amazingly in live, but I couldn’t seem to install it (it recommended installing something else for a permanent OS anyway); and Fedora I got to work but only after doing a lot of extra work. However, I wanted a low-resource system so I figured I’d give Slackware a try.

Which finally leads me to where I’m at now. The install for Salix ran smoothly, but after install, the system won’t boot. It gets to LILO, and then Salix starts up the kernel, until I get this:

[ 7.422388] Kernel panic - not syncing: Fatal exception in interrupt 

Salix is installed in a 155GB partition, there is a Swap partition at 1GB, LILO was installed to the MBR, and the Full install option was chosen for the Salix system.

I’ve looked online, and from what I can see, there are a lot of differing variations and answers to this problem. However, none seemed to match my exact use case.

Any ideas? If anyone knows a way to get a Linux system running on this dang laptop that properly loads the mouse drivers (pointing isn’t iffy like Bunsen Labs and the double-finger click acts as right-click) and Wifi cards, then I will gladly try that. Otherwise, I’d like to at least see if Salix will work since it’s trying to load (which is further than I’ve got with the others).

Here’s the full picture I could get of the screen: full message from boot

## Wifi capabilities disappeared after the most recent kernel update on Ubuntu 16.04 with BCM43142 802.11b/g/n card

I lost WiFi capabilities after a recent Ubuntu 16.04 kernel update.

My network card is BCM43142 802.11b/g/n and I was using

Broadcom 802.11 Linux STA sireless driver source from bcmwl-kernel-source (proprietary)

I tried selecting “Do not use this device” in the Software Updater under “Additional Drivers” and restarted but that did not work.

I also tried downloading and installing the 18.04 binaries with..

sudo dpkg -i bcmwl-kernel-source_6.30.223.271+bdcom-0ubuntu4_amd64.deb 

…and after a restart it didn’t work.

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